Suppose \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of \(\mathcal {C}\) are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of \(\mathcal {C}\) into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and \((n+2)\)-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.
Hinweise
Communicated by Alice Rizzardo.
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1 Introduction
Idempotent completion began with Karoubi’s work [20] on additive categories. It was shown that an additive category embeds into an associated one which is idempotent complete, that is, in which all idempotent morphisms admit a kernel. Particularly nice examples of idempotent complete categories include Krull-Schmidt categories, which can be characterised as idempotent complete additive categories in which each object has a semi-perfect endomorphism ring (see Chen–Ye–Zhang [12, Thm. A.1], Krause [24, Cor. 4.4]). Other examples include the vast class of pre-abelian categories (see e.g. [33, Rem. 2.2]); e.g. a module category, or the category of Banach spaces (over the reals, say).
Suppose \(\mathcal {C}\) is an additive category. The objects of the idempotent completion \(\widetilde{\mathcal {C}}\)of \(\mathcal {C}\) are pairs (X, e), where X is an object of \(\mathcal {C}\) and \(e:X \rightarrow X\) is an idempotent morphism in \(\mathcal {C}\), i.e. \(e^2 = e\). What is particularly nice is that if \(\mathcal {C}\) has a certain kind of structure, then in several cases this induces the same structure on \(\widetilde{\mathcal {C}}\). For example, Karoubi had already shown that the idempotent completion of an additive category is again additive (see [20, (1.2.2)]). Furthermore, it has been shown for the following, amongst other, extrinsic structures that if \(\mathcal {C}\) has such a structure, then so too does \(\widetilde{\mathcal {C}}\):
(i)
triangulated (see Balmer–Schlichting [7, Thm. 1.5]);
Idempotent complete exact and triangulated categories are verifiably important in algebra and algebraic geometry. As a classical example, in Neeman [28] an idempotent complete exact category \(\mathcal {E}\) is needed to give a clean description of the kernel of the localisation functor from the homotopy category of \(\mathcal {E}\) to its derived category. And, more generally, many equivalences only hold up to direct summands, i.e. up to idempotents (see, for example, Orlov [31, Thm. 2.11], or Kalck–Iyama–Wemyss–Yang [21, Thm. 1.1]). Therefore, it is usually helpful to view an algebraic structure as sitting inside its idempotent completion.
The idempotent completion \(\widetilde{\mathcal {C}}\) comes equipped with an inclusion functor given by on objects. Moreover, in several of the cases above it has been shown that this functor is 2-universal in an appropriate sense; see e.g. Proposition 2.8 for a precise formulation. For example, without any assumptions other than additivity, the functor is additive and 2-universal amongst additive functors from \(\mathcal {C}\) to idempotent complete additive categories. On the other hand, if e.g. \(\mathcal {C}\) has an exact structure, then is exact and 2-universal amongst exact functors from \(\mathcal {C}\) to idempotent complete exact categories.
In homological algebra two parallel generalisations have been made from the classical settings of exact and triangulated categories. One of these has been the introduction of extriangulated categories as defined by Nakaoka–Palu [30]. An extriangulated category is a triplet \((\mathcal {C},\mathbb {E},\mathfrak {s})\), where \(\mathcal {C}\) is an additive category, is a biadditive functor to the category of abelian groups, and \(\mathfrak {s}\) is a so-called additive realisation of \(\mathbb {E}\). The realisation \(\mathfrak {s}\) associates to each \(\delta \in \mathbb {E}(Z,X)\) a certain equivalence class of a 3-term complex. As an example, each triangulated category \((\mathcal {C},\Sigma ,{{\triangle }})\), where \(\Sigma \) is a suspension functor and \({{\triangle }}\) is a triangulation, is an extriangulated category. Indeed, one defines the corresponding bifunctor by . See [30, Prop. 3.22] for more details. In addition, each suitable exact category is extriangulated; see [30, Exam. 2.13]. A particular advantage of this theory is that the collection of extriangulated categories is closed under taking extension-closed subcategories. Although an extension-closed subcategory of an exact category is again exact, the same does not hold in general for triangulated categories.
We note here that, importantly, it was shown in [27, Sec. 3.1] that the extriangulated structure on \(\widetilde{\mathcal {C}}\) produced from case (iii) above is compatible with the more classical constructions of (i) and (ii). For instance, given a triangulated category \(\mathcal {C}\), one can equip its idempotent completion \(\widetilde{\mathcal {C}}\) with a triangulation by (i) or with an extriangulation by (iii), but these structures are the same in the sense of [30, Prop. 3.22]. Analogously, (iii) also recovers (ii) if one starts with an extriangulated category that is exact.
Anzeige
Let \(n\geqslant 1\) be an integer. The other aforementioned generalisation in homological algebra has been the development of higher homological algebra. This includes the introduction of n-exact and n-abelian categories by Jasso [19], and \((n+2)\)-angulated categories by Geiss–Keller–Oppermann [14]. Respectively, these generalise exact, abelian and triangulated categories, in that one recovers the classical notions by setting \(n=1\). For instance, an \((n+2)\)-angulated category is a triplet satisfying some axioms, where \(\Sigma \) is still an automorphism of \(\mathcal {C}\), but now consists of a collection of \((n+2)\)-angles each of which has \(n+3\) terms.
The focal point of this paper is on the idempotent completion of an n-exangulated category. These categories were axiomatised by Herschend–Liu–Nakaoka [16], and simultaneously generalise extriangulated, \((n+2)\)-angulated, and suitable n-exact categories (see [16, Sec. 4]). Like an extriangulated category, an n-exangulated category \((\mathcal {C},\mathbb {E},\mathfrak {s})\) consists of an additive category \(\mathcal {C}\), a biadditive functor , and a so-called exact realisation \(\mathfrak {s}\) of \(\mathbb {E}\), which satisfy some axioms (see Sect. 3.1). The realisation \(\mathfrak {s}\) now associates to each a certain equivalence class (see Sect. 3.1)
of an \((n+2)\)-term complex. In this case, the pair is called an \(\mathfrak {s}\)-distinguishedn-exangle. We recall that structure-preserving functors between n-exangulated categories were defined in [10, Def. 2.32]. They are known as n-exangulated functors and they send distinguished n-exangles to distinguished n-exangles.
×
Suppose that \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an n-exangulated category. Let \(\widetilde{\mathcal {C}}\) denote the idempotent completion of \(\mathcal {C}\) as an additive category. We define a biadditive functor as follows. For any pair of objects \((X,e),(Z,e')\in \widetilde{\mathcal {C}}\), we let \(\mathbb {F}((Z,e'),(X,e))\) consist of triplets \((e,\delta ,e')\) where \(\delta \in \mathbb {E}(Z,X)\) such that \(\mathbb {E}(Z,e)(\delta ) = \delta = \mathbb {E}(e',X)(\delta )\). On morphisms \(\mathbb {F}\) is essentially a restriction of \(\mathbb {E}\); see Definition 4.4 for details. Now we define a realisation \(\mathfrak {t}\) of \(\mathbb {F}\). For \((e,\delta ,e') \in \mathbb {F}((Z,e'),(X,e))\), we have that for some \((n+2)\)-term complex with and since \(\mathfrak {s}\) is a realisation of \(\mathbb {E}\). We choose an idempotent morphism of complexes, such that and ; see Corollary 4.13. Lastly, we set \(\mathfrak {t}((e,\delta ,e'))\) to be the equivalence class of the complex
in \(\widetilde{\mathcal {C}}\). We say that an n-exangulated category is idempotent complete if its underlying additive category is (see Definition 4.31).
×
Theorem A
(Theorem 4.32, Theorem 4.39) The triplet \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is an idempotent complete n-exangulated category. Furthermore, the inclusion functor extends to an n-exangulated functor , which is 2-universal among n-exangulated functors from \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) to idempotent complete n-exangulated categories.
An n-exact category\((\mathcal {C}, \mathcal {X})\) (see [19, Def. 4.2]) induces an n-exangulated category \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) if, for each pair of objects \(A,C\in \mathcal {C}\), the collection of n-extensions of C by A forms a set; see [16, Prop. 4.34]. As in [23, Def. 4.6], we say that an n-exangulated category \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) is n-exact if its n-exangulated structure arises in this way. Combining Theorem A with [23, Cor. 4.12], we deduce the following.
Corollary B
(Corollary 4.34) If \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an n-exangulated category that is n-exact, then the idempotent completion \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is n-exact.
We explain in Remark 4.40 how Theorem A unifies the constructions in cases (i)–(iv) above. Furthermore, we comment on some obstacles faced in proving the n-exangulated case in Remark 4.41.
From Theorem A we deduce the following corollary, giving a way to produce Krull-Schmidt n-exangulated categories.
Corollary C
(Corollary 4.33) If each object in \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) has a semi-perfect endomorphism ring, then the idempotent completion \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is a Krull-Schmidt n-exangulated category.
Finally, we note that analogues of Theorem A and Corollary B are shown for the weak idempotent completion in Sect. 5. The importance of being weakly idempotent complete for extriangulated categories was very recently demonstrated in [23, Prop. 2.7]. It turns out that for an extriangulated category, the underlying category being weakly idempotent complete is equivalent to the condition (WIC) defined in [30, Cond. 5.8]. Moreover, (WIC) is a key assumption in many results on extriangulated categories, e.g. [30, §§5–7], [17, §3], Zhao–Zhu–Zhuang [36]. We remark that the analogue of (WIC) for n-exangulated categories is automatic if \(n\geqslant 2\), but it is not equivalent to the weak idempotent completeness of the underlying category; see [23, Thm. B] for more details.
2 On the Splitting of Idempotents
In this section we recall some key definitions regarding idempotents and idempotent completions of categories. We focus on the idempotent completion of an additive category in Sect. 2.1 and on the weak idempotent completion in Sect. 2.2. Throughout this section, we let \(\mathcal {A}\) denote an additive category. For a more in-depth treatment, we refer the reader to [11, Secs. 6–7].
2.1 Idempotent Completion
Recall that by an idempotent (in \(\mathcal {A}\)) we mean a morphism \(e:X\rightarrow X\) satisfying \(e^2 = e\) for some object \(X\in \mathcal {A}\).
[6, Defs. 6.5.1, 6.5.3] An idempotent \(e :X \rightarrow X\) in \(\mathcal {A}\) is said to split if there exist morphisms \(r :X \rightarrow Y\) and \(s :Y \rightarrow X\), such that \(sr=e\) and . The category \(\mathcal {A}\) is idempotent complete, or has split idempotents, if every idempotent in \(\mathcal {A}\) splits.
If \(\mathcal {A}\) has split idempotents and \(e:X \rightarrow X\) is an idempotent in \(\mathcal {A}\), then the object X admits a direct sum decomposition (see e.g. Auslander [3, p. 188]). In particular, the idempotent e and its counterpart each admit a kernel. Idempotent complete additive categories can be characterised by such a criterion and its dual.
Proposition 2.2
[6, Prop. 6.5.4] An additive category is idempotent complete if and only if every idempotent admits a kernel, if and only if every idempotent admits a cokernel.
From this point of view, idempotent complete categories sit between additive categories and pre-abelian categories, the latter being additive categories in which every morphism admits a kernel and a cokernel; see for example Bucur–Deleanu [4, §5.4].
Every additive category can be viewed as a full subcategory of an idempotent complete one. This goes back to Karoubi [20, Sec. 1.2], so the idempotent completion of \(\mathcal {A}\) is also often referred to as the Karoubi envelope of\(\mathcal {A}\).
Definition 2.3
The idempotent completion\(\widetilde{\mathcal {A}}\)of\(\mathcal {A}\) is the category defined as follows. Objects of \(\widetilde{\mathcal {A}}\) are pairs (X, e), where X is an object of \(\mathcal {A}\) and is idempotent. For objects \((X,e), (Y,e') \in \hbox {obj}\widetilde{\mathcal {A}}\), a morphism from (X, e) to \((Y,e')\) is a triplet \((e',r,e)\), where \(r \in \mathcal {A}(X,Y)\) satisfies
$$\begin{aligned} re = r = e'r \end{aligned}$$
in \(\mathcal {A}\). Composition of morphisms is defined by
whenever \((e',r,e)\in \widetilde{\mathcal {A}}((X,e), (Y, e'))\) and \((e'',s,e')\in \widetilde{\mathcal {A}}((Y,e'), (Z, e''))\). The identity of an object \((X,e)\in \hbox {obj}\widetilde{\mathcal {A}}\) will be denoted and is the morphism (e, e, e).
A morphism \((e',r,e) :(X,e) \rightarrow (Y,e')\) in the idempotent completion \(\widetilde{\mathcal {A}}\) of \(\mathcal {A}\) is usually denoted more simply as r; see e.g. [7, Def. 1.2] and [11, Rem. 6.3]. However, for precision in Sects. 4–5, we use triplets for morphisms in \(\widetilde{\mathcal {A}}\) so that we can easily distinguish morphisms in \(\mathcal {A}\) from morphisms in its idempotent completion. Our choice of notation also has the added benefit of keeping track of the (co)domain of a morphism in \(\widetilde{\mathcal {A}}\). This becomes important later when different morphisms in \(\widetilde{\mathcal {A}}\) have the same underlying morphism; see Notation 4.37.
By a functor we always mean a covariant functor. The inclusion functor is defined as follows. An object \(X \in \hbox {obj}\mathcal {A}\) is sent to and a morphism \(r \in \mathcal {A}(X,Y)\) is mapped to .
Lemma 2.4
If is a split idempotent, with a splitting \(e=sr\) where \(r :X \rightarrow Y\) and \(s :Y \rightarrow X\), then .
Proof
We have and . Hence, there are morphisms and in \(\widetilde{\mathcal {A}}\) with and . Hence, \(\tilde{r}\) and \(\tilde{s}\) are mutually inverse isomorphisms in \(\widetilde{\mathcal {A}}\). \(\square \)
If \(\mathcal {A}\) is an idempotent complete category, then the functor is an equivalence of categories; see e.g. [11, Rem. 6.5]. But more generally we have the following.
Proposition 2.5
[11, Rem. 6.3] The idempotent completion \(\widetilde{\mathcal {A}}\) is an idempotent complete additive category with biproduct given by \((X, e) \oplus (Y, e') = (X \oplus Y, e \oplus e')\). The inclusion functor is fully faithful and additive.
Remark 2.6
Let (X, e) be an arbitrary object of \(\widetilde{\mathcal {A}}\). Then (X, e) is a direct summand of . Indeed, there is an isomorphism . The canonical inclusion of (X, e) into is given by the morphism , and the projection of onto (X, e) by . Similarly for .
The functor is 2-universal in some sense; see Proposition 2.8. For this we recall the notion of whiskering a natural transformation by a functor. We will use Hebrew letters (e.g. (beth), (tsadi), (daleth), (mem)) for natural transformations. Suppose \(\mathcal {B},\mathcal {C},\mathcal {D}\) are categories and that we have a diagram
where \(\mathscr {F},\mathscr {G},\mathscr {H}\) are functors and is a natural transformation.
×
Definition 2.7
The whiskering of\(\mathscr {F}\)and is the natural transformation defined by for each \(X\in \mathcal {B}\).
The next proposition explains the 2-universal property satisfied by .
Proposition 2.8
[11, Prop. 6.10] For any additive functor \(\mathscr {F}:\mathcal {A}\rightarrow \mathcal {B}\) with \(\mathcal {B}\) idempotent complete:
(i)
there is an additive functor \(\mathscr {E}:\widetilde{\mathcal {A}} \rightarrow \mathcal {B}\) and a natural isomorphism and, in addition,
(ii)
for any functor \(\mathscr {G}:\widetilde{\mathcal {A}} \rightarrow \mathcal {B}\) and any natural transformation , there exists a unique natural transformation with .
2.2 Weak Idempotent Completion
A weaker notion than being idempotent complete is that of being weakly idempotent complete. This was introduced in the context of exact categories by Thomason–Trobaugh [34, Axiom A.5.1]. It is, however, a property of the underlying additive category and gives rise to the following definition.
Definition 2.9
[11, Def. 7.2] An additive category is weakly idempotent complete if every retraction has a kernel.
Definition 2.9 is actually self-dual. Indeed, in an additive category, every retraction has a kernel if and only if every section has a cokernel; see e.g. [11, Lem. 7.1].
If \(r :X \rightarrow Y\) is a retraction in \(\mathcal {A}\), with corresponding section \(s :Y \rightarrow X\), and r admits a kernel k, then the split idempotent also has kernel k. Conversely, if \(e:X\rightarrow X\) is a split idempotent, with splitting given by \(e=sr\) where \(r :X \rightarrow Y\), then a kernel of e is also a kernel of r. Therefore, weakly idempotent complete categories are those additive categories in which split idempotents admit kernels, in contrast to idempotent complete categories in which all idempotents admit kernels (see Proposition 2.2).
Definition 2.10
The weak idempotent completion \(\widehat{\mathcal {A}}\)of\(\mathcal {A}\) is the full subcategory of \(\widetilde{\mathcal {A}}\) consisting of all objects \((X, e)\in \widetilde{\mathcal {A}}\) such that is a split idempotent in \(\mathcal {A}\).
Remark 2.11
We note that Definition 2.10 above differs slightly from the definition of the weak idempotent completion of \(\mathcal {A}\) suggested in [11, Rem. 7.8]. If, as in [11], we ask that objects of \(\widehat{\mathcal {A}}\) are pairs (X, e) where \(e:X\rightarrow X\) splits, then \(\widehat{\mathcal {A}}\) is equivalent to \(\mathcal {A}\). Indeed, if \(sr=e\) and , where \(r:X\rightarrow Y\) and \(s:Y\rightarrow X\), then in \(\widetilde{\mathcal {A}}\) by Lemma 2.4. That is, we have not added any objects that are not already isomorphic to some object of . On the other hand, if we take objects in \(\widehat{\mathcal {A}}\) to be pairs (X, e) where splits (as in Definition 2.10), then we have in \(\widehat{\mathcal {A}}\), where and , where \(r':X\rightarrow Y'\) and \(s':Y'\rightarrow X\). In this case, since in \(\widetilde{\mathcal {A}}\), we see that a “complementary” summand of in has been added. This discrepancy has been noticed previously; see e.g. Henrard–van Roosmalen [18, Prop. A.11].
It follows that \(\widehat{\mathcal {A}}\) is an additive subcategory of \(\widetilde{\mathcal {A}}\) and that it is weakly idempotent complete; see e.g. [11, Rem. 7.8] or [18, Sec. A.2]. From this observation, we immediately have the next lemma.
Lemma 2.12
Suppose \(\widetilde{X}, \widetilde{Y}, \widetilde{Z} \in \widetilde{\mathcal {A}}\) with \(\widetilde{X} \oplus \widetilde{Y} \cong \widetilde{Z}\). Then any two of \(\widetilde{X}, \widetilde{Y}, \widetilde{Z}\) being isomorphic to objects in \(\widehat{\mathcal {A}}\) implies that the third object is also isomorphic to an object in \(\widehat{\mathcal {A}}\).
Analogously to the construction in Sect. 2.1, there is an inclusion functor , given by on objects, which is 2-universal among additive functors from \(\mathcal {A}\) to weakly idempotent complete categories; see e.g. [28, Rem. 1.12] or [11, Rem. 7.8].
Proposition 2.13
For any additive functor \(\mathscr {F}:\mathcal {A}\rightarrow \mathcal {B}\) with \(\mathcal {B}\) weakly idempotent complete:
(i)
there is an additive functor \(\mathscr {E}:\widehat{\mathcal {A}} \rightarrow \mathcal {B}\) and a natural isomorphism ; and, in addition,
(ii)
for any additive functor \(\mathscr {G}:\widehat{\mathcal {A}} \rightarrow \mathcal {B}\) and any natural transformation , there exists a unique natural transformation with .
Let denote the inclusion functor of the subcategory \(\widehat{\mathcal {A}}\) into \(\widetilde{\mathcal {A}}\). The functor factors through as . An additive functor \(\mathscr {F}:\widehat{\mathcal {A}} \rightarrow \mathcal {B}\) to a weakly idempotent complete category \(\mathcal {B}\) is determined up to unique natural isomorphism by its behaviour on the image of \(\mathcal {A}\) in \(\widehat{\mathcal {A}}\); similarly, a natural transformation of additive functors \(\widehat{\mathcal {A}}\rightarrow \mathcal {B}\) is also completely determined by its action on ; see [11, Rems. 6.7, 6.9].
Remark 2.14
In [11, Rem. 7.9], it is remarked that there is a subtle set-theoretic issue regarding the existence of the weak idempotent completion of an additive category. Let NBG denote von Neumann-Bernays-Gödel class theory (see Fraenkel–Bar-Hillel–Levy [13, p. 128]), and let (AGC) denote the Axiom of Global Choice [13, p. 133]. The combination NBG + (AGC) is a conservative extension of ZFC [13, p. 131–132, 134]. If one chooses an appropriate class theory to work with, such as NBG + (AGC), then the weak idempotent completion always exists as a category. This would follow from the Axiom of Predicative Comprehension for Classes (see [13, p. 123]); this is also known as the Axiom of Separation (e.g. Smullyan–Fitting [32, p. 15]). Furthermore, a priori it is not clear to the authors if Proposition 2.8 and 2.13 follow in an arbitrary setting without (AGC). This is because in showing that, for example, an additive functor \(\mathscr {F}:\widetilde{\mathcal {A}} \rightarrow \mathcal {B}\), where \(\mathcal {B}\) is idempotent complete, is determined by its values on , one must choose a kernel and an image of the idempotent \(\mathscr {F}(e)\) for each idempotent e in \(\mathcal {A}\).
3 n-Exangulated Categories, Functors and Natural Transformations
Let \(n\geqslant 1\) be an integer. In this section we recall the theory of n-exangulated categories established in [16], n-exangulated functors as defined in [10], and n-exangulated natural transformations as recently introduced in [9]. We also use this opportunity to set up some notation.
3.1 n-Exangulated Categories
The definitions in this subsection and more details can be found in [16, Sec. 2]. For this subsection, suppose that \(\mathcal {C}\) is an additive category and that is a biadditive functor.
Let A, C be objects in \(\mathcal {C}\). We denote by the identity element of the abelian group \(\mathbb {E}(C,A)\). Suppose \(\delta \in \mathbb {E}(C,A)\) and that \(a:A\rightarrow B\) and \(d:D\rightarrow C\) are morphisms in \(\mathcal {C}\). We put and . Since \(\mathbb {E}\) is a bifunctor, we have that .
An \(\mathbb {E}\)-extension is an element \(\delta \in \mathbb {E}(C,A)\) for some \(A,C\in \mathcal {C}\). A morphism of\(\mathbb {E}\)-extensions from \(\delta \in \mathbb {E}(C,A)\) to \(\rho \in \mathbb {E}(D,B)\) is given by a pair (a, c) of morphisms \(a:A\rightarrow B\) and \(c:C\rightarrow D\) in \(\mathcal {C}\) such that .
Let be a product and be a coproduct in \(\mathcal {C}\), and let \({\delta \in \mathbb {E}(C,A)}\) and \(\rho \in \mathbb {E}(D,B)\) be \(\mathbb {E}\)-extensions. The direct sum of\(\delta \)and\(\rho \) is the unique \(\mathbb {E}\)-extension \(\delta \oplus \rho \in \mathbb {E}(C\oplus D,A\oplus B)\) such that the following equations hold.
From the Yoneda Lemma, each \(\mathbb {E}\)-extension \(\delta \in \mathbb {E}(C,A)\) induces two natural transformations. The first is given by for all objects \(B\in \mathcal {C}\) and all morphisms \(a:A\rightarrow B\). The second is and defined by for all objects \(D\in \mathcal {C}\) and all morphisms \(d:D\rightarrow C\).
Let \(\textsf {\textrm{Ch}}({\mathcal {C}})\) be the category of complexes in \(\mathcal {C}\). Its full subcategory consisting of complexes concentrated in degrees \(0,1,\ldots , n,n+1\) is denoted . If , we depict as
omitting the trails of zeroes at each end.
×
Definition 3.1
Let be complexes, and suppose that and are \(\mathbb {E}\)-extensions.
(i)
The pair is known as an \(\mathbb {E}\)-attached complex if and . An \(\mathbb {E}\)-attached complex is called an n-exangle (for \((\mathcal {C},\mathbb {E})\)) if, further, the sequences
of functors are exact.
×
(ii)
A morphismof\(\mathbb {E}\)-attached complexes is given by a morphism such that . Such an is called a morphism ofn-exangles if and are both n-exangles.
(iii)
The direct sum of the\(\mathbb {E}\)-attached complexes (or then-exangles) and is the pair .
From the definition above, one can form the additive category of \(\mathbb {E}\)-attached complexes, and its additive full subcategory of n-exangles.
Given a pair of objects \(A,C\in \mathcal {C}\), we define a subcategory of in the following way. An object is an object of that satisfies and . For , a morphism is a morphism with and . Note that this implies is not necessarily a full subcategory of , nor necessarily additive.
Let be complexes. Two morphisms in are said to be homotopic if they are homotopic in the standard sense viewed as morphisms in . This induces an equivalence relation \(\sim \) on . We define as the category with the same objects as and with .
A morphism is called a homotopy equivalence if its image in the category is an isomorphism. In this case, and are said to be homotopy equivalent. The isomorphism class of in (equivalently, its homotopy class in ) is denoted . Since the (usual) homotopy class of in \(\textsf {\textrm{Ch}}({\mathcal {C}})\) may differ from its homotopy class in , we reserve the notation specifically for its isomorphism class in .
Notation 3.2
For \(X \in \mathcal {C}\) and \(i \in \{ 0, \dots , n\}\), we denote by the object in given by for \(j = i, i+1\) and for \(0 \leqslant j \leqslant i-1\) and \(i+2 \leqslant j \leqslant n+1\), as well as .
Definition 3.3
Let \(\mathfrak {s}\) be an assignment that, for each pair of objects \(A,C\in \mathcal {C}\) and each \(\mathbb {E}\)-extension \({\delta \in \mathbb {E}(C,A)}\), associates to \(\delta \) an isomorphism class in . The correspondence \(\mathfrak {s}\) is called an exact realisation of \(\mathbb {E}\) if it satisfies the following conditions.
(R0)
For any morphism \((a,c):\delta \rightarrow \rho \) of \(\mathbb {E}\)-extensions with \(\delta \in \mathbb {E}(C,A)\), \(\rho \in \mathbb {E}(D,B)\), and , there exists such that \(f_{0}=a\) and . In this setting, we say that realises\(\delta \) and is a lift of (a, c).
(R1)
If , then is an n-exangle.
(R2)
For each object \(A\in \mathcal {C}\), we have and .
In case \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\) and the following terminology is used. The morphism is said to be an \(\mathfrak {s}\)-inflation and the morphism an \(\mathfrak {s}\)-deflation. The pair is known as an \(\mathfrak {s}\)-distinguishedn-exangle.
Suppose \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\) and . We will often use the diagram
to express that is an \(\mathfrak {s}\)-distinguished n-exangle. If we also have that and is a morphism of n-exangles, then we call a morphism of\(\mathfrak {s}\)-distinguishedn-exangles and we depict this by the following commutative diagram.
×
×
We need one last definition before being able to define an n-exangulated category.
Definition 3.4
Suppose is a morphism in , such that for some . The mapping cone of is the complex
with , , and for \(i\in \{1,\ldots ,n-1\}\).
×
We are in position to state the main definition of this subsection.
Definition 3.5
An n-exangulated category is a triplet \((\mathcal {C},\mathbb {E},\mathfrak {s})\), consisting of an additive category \(\mathcal {C}\), a biadditive functor and an exact realisation \(\mathfrak {s}\) of \(\mathbb {E}\), such that the following conditions are met.
(\(\hbox {EA1}\))
The collection of \(\mathfrak {s}\)-inflations is closed under composition. Dually, the collection of \(\mathfrak {s}\)-deflations is closed under composition.
(\(\hbox {EA2}\))
Suppose \(\delta \in \mathbb {E}(D,A)\) and \(c\in \mathcal {C}(C,D)\). If and , then there exists a morphism lifting , such that . In this case, the morphism is called a good lift of.
The dual of (EA2).
Notice that the definition of an n-exangulated category is self-dual. In particular, the dual statements of several results in Sects. 4–5 are used without proof.
3.2 n-Exangulated Functors and Natural Transformations
In order to show that the canonical functor from an n-exangulated category \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to its idempotent completion is 2-universal among structure-preserving functors from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to idempotent complete n-exangulated categories, we will need the notion of a morphism of n-exangulated categories and that of a morphism between such morphisms.
For this subsection, suppose \((\mathcal {C},\mathbb {E},\mathfrak {s})\), \((\mathcal {C}',\mathbb {E}',\mathfrak {s}')\) and \((\mathcal {C}'',\mathbb {E}'',\mathfrak {s}'')\) are n-exangulated categories. If \(\mathscr {F}:\mathcal {C}\rightarrow \mathcal {C}'\) is an additive functor, then it induces several other additive functors, e.g. and obvious restrictions thereof. These are all defined in the usual way. However, by abuse of notation, we simply write \(\mathscr {F}\) for each of these.
Definition 3.6
[10, Def. 2.32] Suppose that \(\mathscr {F}:\mathcal {C}\rightarrow \mathcal {C}'\) is an additive functor and that \( \Gamma :\mathbb {E}(-,-) \Rightarrow \mathbb {E}'(\mathscr {F}-, \mathscr {F}-) \) is a natural transformation of functors . The pair \((\mathscr {F},\Gamma ) :(\mathcal {C},\mathbb {E},\mathfrak {s}) \rightarrow (\mathcal {C}',\mathbb {E}',\mathfrak {s}')\) is called an n-exangulated functor if, for all \(A,C\in \mathcal {C}\) and each \(\delta \in \mathbb {E}(A,C)\), we have that whenever .
If we have a sequence of n-exangulated functors, then the composite of \((\mathscr {F},\Gamma )\) and \((\mathscr {L},\Phi )\) is defined to be
This is an n-exangulated functor \((\mathcal {C},\mathbb {E},\mathfrak {s})\rightarrow (\mathcal {C}'',\mathbb {E}'',\mathfrak {s}'')\); see [9, Lem. 3.19(ii)].
The next result implies that n-exangulated functors preserve finite direct sum decompositions of distinguished n-exangles. It will be used in the main result of Sect. 4.5.
Proposition 3.7
Let \(\mathscr {F}:\mathcal {C}\rightarrow \mathcal {C}'\) be an additive functor and \({\Gamma :\mathbb {E}(-,-) \Rightarrow \mathbb {E}'(\mathscr {F}-,\mathscr {F}-)}\) a natural transformation. Suppose \(\delta \in \mathbb {E}(C,A)\) and \(\rho \in \mathbb {E}(D,B)\) are \(\mathbb {E}\)-extensions, and and are \(\mathfrak {s}\)-distinguished.
(i)
If is a morphism of \(\mathbb {E}\)-attached complexes, then the induced morphism is a morphism of \(\mathbb {E}'\)-attached complexes.
(ii)
We have as \(\mathbb {E}'\)-attached complexes.
Proof
(i) Note that since \(\Gamma \) is natural and is an \(\mathbb {E}\)-attached complex. Similar computations show that both and are \(\mathbb {E}'\)-attached complexes. As is a morphism of complexes, it suffices to prove
This follows immediately from and the naturality of \(\Gamma \).
(ii) This follows from applying (i) to the morphisms in the appropriate biproduct diagram of \(\mathbb {E}\)-attached complexes. \(\square \)
Lastly, we recall the notion of a morphism of n-exangulated functors. The extriangulated version was defined in Nakaoka–Ogawa–Sakai [29, Def. 2.11(3)].
Definition 3.8
[9, Def. 4.1] Suppose \((\mathscr {F},\Gamma ), (\mathscr {G},\Lambda ):(\mathcal {C},\mathbb {E},\mathfrak {s})\rightarrow (\mathcal {C}',\mathbb {E}',\mathfrak {s}')\) are n-exangulated functors. A natural transformation of functors is said to be n-exangulated if, for all \(A,C\in \mathcal {C}\) and each \(\delta \in \mathbb {E}(C,A)\), we have
(3.1)
We denote this by . In addition, if has an n-exangulated inverse, then it is called an n-exangulated natural isomorphism. It is straightforward to check that has an n-exangulated inverse if and only if is an isomorphism for each \(X\in \mathcal {C}\).
4 The Idempotent Completion of an n-Exangulated Category
Throughout this section we work with the following setup.
Setup 4.1
Let \(n\geqslant 1\) be an integer. Let \((\mathcal {C},\mathbb {E},\mathfrak {s})\) be an n-exangulated category. We denote by the inclusion of the category \(\mathcal {C}\) into its idempotent completion \(\widetilde{\mathcal {C}}\); see Sect. 2.
In this section, we will construct a biadditive functor (see Sect. 4.1) and an exact realisation \(\mathfrak {t}\) of \(\mathbb {F}\) (see Sect. 4.2), and then show that \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) is an n-exangulated category (see Sects. 4.3–4.5). For \(n=1\), we recover the main results of [27]. First, we establish some notation to help our exposition.
Notation 4.2
We reserve notation with a tilde for objects and morphisms in \(\widetilde{\mathcal {C}}\).
(i)
If \(\widetilde{X}\in \widetilde{\mathcal {A}}\) is some object, then we will denote the identity morphism of \(\widetilde{X}\) by . Recall from Definition 2.3 that the identity of an object \((X,e)\in \widetilde{\mathcal {A}}\) is .
(ii)
Given a morphism \((e',r,e)\in \widetilde{\mathcal {C}}((X,e),(Y,e'))\), we call \(r:X\rightarrow Y\) the underlying morphism of \((e',r,e)\).
(iii)
Suppose \((X,e), (Y,e') \in \widetilde{\mathcal {C}}\) and \(r \in \mathcal {C}(X, Y)\) with \(e' r = r = r e\). Then there is a unique morphism \(\tilde{r} \in \widetilde{\mathcal {C}}((X,e), (Y,e'))\) with underlying morphism r. This morphism \(\tilde{r}\) is the triplet \((e', r, e)\). Moreover, we will use this notation specifically for this correspondence. That is, we write \(\tilde{s}:(X,e) \rightarrow (Y,e')\) is a morphism in \(\widetilde{\mathcal {C}}\) if and only if we implicitly mean that the underlying morphism of \(\tilde{s}\) is denoted s, i.e. we have \(\tilde{s} = (e',s,e)\).
Remark 4.3
By Notation 4.2(iii), two morphisms \(\tilde{r}, \tilde{s} \in \widetilde{\mathcal {C}} ((X,e), (Y,e'))\) are equal if and only if their underlying morphisms r and s, respectively, are equal in \(\mathcal {C}\). Thus, for all objects \(\widetilde{X},\widetilde{Y}\in \widetilde{\mathcal {C}}\), removing the tilde from morphisms in \(\widetilde{\mathcal {C}}(\widetilde{X},\widetilde{Y})\) defines an injective abelian group homomorphism \(\widetilde{\mathcal {C}}(\widetilde{X},\widetilde{Y}) \rightarrow \mathcal {C}(X,Y)\). In particular, a diagram in \(\widetilde{\mathcal {C}}\) commutes if and only if its diagram of underlying morphisms commutes.
4.1 Defining the Biadditive Functor \(\mathbb {F}\)
The following construction is the higher version of the one given in [27, Sec. 3.1] for extriangulated categories.
Definition 4.4
We define a functor as follows. For objects and in \(\widetilde{\mathcal {C}}\), we put
For morphisms in \(\widetilde{\mathcal {C}}\), we define
The assignment \(\mathbb {F}\) on morphisms takes values where claimed due to the following. For morphisms and , and an \(\mathbb {F}\)-extension , we have
Therefore, lies in . It is then straightforward to verify that \(\mathbb {F}\) is indeed a functor.
(ii)
The set is an abelian group by defining
for . The additive identity element of is . The inverse of is . Notice that we get an abelian group monomorphism:
This homomorphism plays a role later in the proof of Theorem 4.39.
(iii)
It follows from the definition of \(\mathbb {F}\) that it is biadditive since \(\mathbb {E}\) is.
(iv)
Given , the pair is a morphism of \(\mathbb {E}\)-extensions \(\delta \rightarrow \delta \). Indeed, we have that from Definition 4.4.
Notation 4.6
As for objects and morphisms in \(\widetilde{\mathcal {C}}\), we use tilde notation for \(\mathbb {F}\)-extensions, which gives us a way to pass back to \(\mathbb {E}\)-extensions.
(i)
We will denote an \(\mathbb {F}\)-extension of the form by \({\tilde{\delta }}\). We call the underlying\(\mathbb {E}\)-extension of\({\tilde{\delta }}\).
(ii)
For and with , there is a unique \(\mathbb {F}\)-extension with underlying \(\mathbb {E}\)-extension \(\delta \). This \(\mathbb {F}\)-extension is . Again, we use this instance of the tilde notation for this correspondence: we write if and only if the underlying \(\mathbb {E}\)-extension of \({\tilde{\rho }}\) is \(\rho \), i.e. .
Remark 4.7
Analogously to our observations in Remark 4.3, we note that by Notation 4.6(ii) any two \(\mathbb {F}\)-extensions are equal if and only if their underlying \(\mathbb {E}\)-extensions are equal. Hence, removing the tilde from \(\mathbb {F}\)-extensions defines an injective abelian group homomorphism \(\mathbb {F}((Y,e'), (X,e)) \rightarrow \mathbb {E}(Y,X)\) for \((X,e), (Y,e') \in \widetilde{\mathcal {C}}\).
4.2 Defining the Realisation \(\mathfrak {t}\)
To define an exact realisation \(\mathfrak {t}\) of the functor \(\mathbb {F}\) defined in Sect. 4.1, given a morphism of extensions consisting of two idempotents, we will need to lift this morphism to an \((n+2)\)-tuple of idempotents. That is, we require a higher version of the idempotent lifting trick (see [27, Lem. 3.5] and [7, Lem. 1.13]). This turns out to be quite non-trivial and requires an abstraction of the case when \(n=1\) in order to understand the mechanics of why this trick is successful.
We start with two lemmas related to the polynomial ring \(\mathbb {Z}[x]\). Recall that \(\mathbb {Z}[x]\) has the universal property that for any (unital, associative) ring R and any element \(r \in R\) there is a unique (identity preserving) ring homomorphism \(\varphi _r :\mathbb {Z}[x] \rightarrow R\) with \(\varphi _r(x) = r\). For \(p=p(x) \in \mathbb {Z}[x]\), we denote \(\varphi _r(p)\) by p(r) as is usual.
Lemma 4.8
For each \(m \in \mathbb {N}\), the ideals \((x^m) = {(x)}^m\) and \(((x-1)^m) = {(x-1)}^m\) of \(\mathbb {Z}[x]\) are coprime.
Proof
The ideals \(\sqrt{{(x)}^m} = (x)\) and \(\sqrt{{(x-1)}^m} = (x-1)\) are coprime in \(\mathbb {Z}[x]\). Hence, \((x^m)\) and \(( (x-1)^m )\) are also coprime by Atiyah–MacDonald [2, Prop. 1.16]. \(\square \)
Lemma 4.9
For each \(m \in \mathbb {N}_{\geqslant 1}\), there is a polynomial \(p_m \in (x^m) \unlhd \mathbb {Z}[x]\), such that for every (unital, associative) ring R we have:
(i)
\(p_m(e) = e\) for each idempotent \(e \in R\); and
(ii)
the element \(p_m(r) \in R\) is an idempotent for each \(r \in R\) satisfying \((r^2-r)^m = 0\).
Proof
Fix an integer \(m \geqslant 1\). By Lemma 4.8, we can write \(1 = x^m p_m' + (x-1)^m q_m'\) for some polynomials \(p_m'\) and \(q_m'\) in \(\mathbb {Z}[x]\). We set \(p_m :=x^m p_m'\).
Let R be a ring. For any idempotent \(e \in R\), evaluating \(x = x^{m+1} p_m' + x(x-1)^m q_m'\) at e and using \(e(e-1) = 0\) yields \(e = e^{m+1} p_m'(e) = e^m p_m'(e) = p_m(e)\), proving (i).
Now suppose \(r \in R\) is an element with \((r^2-r)^m = 0\). Evaluation of
at r shows \(p_m(r)^2 = p_m(r)\) since \((r^2-r)^m = 0\), which finishes the proof. \(\square \)
The following is an abstract formulation of [27, Lem. 3.5] and [7, Lem. 1.13].
Lemma 4.10
Let be a complex in an additive category \(\mathcal {A}\) and suppose is a weak cokernel of . Suppose is a morphism of complexes with and both idempotent. Then there exists a morphism , such that the following hold.
(i)
The triplet is a morphism of complexes.
(ii)
The element is idempotent and satisfies .
(iii)
The triplet is an idempotent morphism of complexes.
(iv)
If is a homotopy of morphisms , then the pair yields a homotopy .
Proof
Choose a polynomial as obtained in Lemma 4.9. Define \(q :=xp'_{2}\) and set . We show this morphism satisfies the claims in the statement. For this, we will make use of the following. Let \(p=p(x)\in \mathbb {Z}[x]\) be any polynomial. Since is a morphism of complexes, we have that is also a morphism of complexes, i.e. the diagram
commutes.
×
(i) Note that , where the last equality follows from Lemma 4.9(i). Similarly, . Thus, using \(p=q\) in the commutative diagram (4.1) shows that is a morphism of complexes.
(ii) Since \(f_1' = q(f_1)\) is a polynomial in \(f_1\), we immediately have that . Furthermore, we see that . Thus, to show that is idempotent, it is enough to show that \((f_1^2 - f_1)^2 = 0\) by Lemma 4.9(ii). Let \(r(x) = x^2 -x\). We see that and vanish as and are idempotents. Therefore, by choosing \(p = r\) in (4.1) we have and so there is with , because is a weak cokernel of . This implies as , and hence is idempotent.
(iii) Note that is a morphism of complexes using in (4.1).
(iv) Suppose is a homotopy. Then we see that
Hence, is a null homotopy as desired. \(\square \)
Remark 4.11
Let \(p'_2 = -2x + 3\) and \(q'_2 = 2x + 1\). Then indeed \(1 = x^2p'_2 + (x-1)^2q'_2\). Hence, is a possible choice for \(m=2\) in Lemma 4.9. Letting \(h = x^2 - x\) and \(i = x\), we see that . Then the idempotent obtained in Lemma 4.10 is the idempotent obtained through the idempotent lifting trick in [27, Lem. 3.5].
Lemma 4.12
Suppose is an \(\mathfrak {s}\)-distinguished n-exangle and is an idempotent with . Then can be extended to a null homotopic, idempotent morphism with for \(2 \leqslant i \leqslant n+1\). Further, the null homotopy of \(e_{\bullet }\) can be chosen to be of the shape .
Proof
We have so is a morphism of \(\mathbb {E}\)-extensions. The solid morphisms of the diagram
clearly commute, so we need to find a morphism making the two leftmost squares commute. Since is an \(\mathfrak {s}\)-distinguished n-exangle, there is an exact sequence
The morphism is in the kernel of as . Therefore, there exists with . If we put , then is morphism of \(\mathfrak {s}\)-distinguished n-exangles and is a homotopy. By Lemma 4.10, using that and 0 are idempotents, there is an idempotent , such that is an idempotent morphism of complexes and that is a homotopy. Finally, is a morphism of \(\mathfrak {s}\)-distinguished n-exangles since . \(\square \)
×
×
Corollary 4.13
Suppose and that is an \(\mathfrak {s}\)-distinguished n-exangle. The morphism of \(\mathbb {E}\)-extensions has a lift that is idempotent and satisfies for all \(2 \leqslant i \leqslant n -1\), such that there is a homotopy .
Proof
Define and , we have and so . Therefore, by Lemma 4.12 we can extend of \(\mathfrak {s}\)-distinguished n-exangles with \(e'_{i} = 0\) for \(i\in \{ 2,\ldots ,n+1 \}\), having a homotopy . Similarly, by the dual of Lemma 4.12, we can extend \(e''_{n+1}\) to an idempotent morphism with \(e''_{i} = 0\) for \(i\in \{ 0,\ldots ,n-1 \}\), such that there is a homotopy . Consider the morphism and hence is a homotopy.
If \(n = 1\), then and is a homotopy. Lemma 4.10 yields an idempotent morphism and a homotopy . Then and are the desired idempotent morphism and homotopy, respectively.
If \(n \geqslant 2\), then the compositions \(e'_{\bullet }e''_{\bullet } \) and \(e''_{\bullet } e'_{\bullet }\) are zero. This implies that is idempotent. Hence, and are the desired idempotent morphism and homotopy, respectively. \(\square \)
The following simple lemma will be used several times.
Lemma 4.14
Suppose that \((X,e), (Y,e')\) are objects in \(\widetilde{\mathcal {C}}\) and \(r:X\rightarrow Y\) is a morphism in \(\mathcal {C}\). Setting \(s :=e're\) yields a morphism \(\tilde{s} = (e',s,e) :(X,e)\rightarrow (Y,e')\) in \(\widetilde{\mathcal {C}}\).
The previous result allows us to view a complex in \(\mathcal {C}\) that is equipped with an idempotent endomorphism as a complex in the idempotent completion \(\widetilde{\mathcal {C}}\), as follows.
Definition 4.15
Suppose is a complex in \(\mathcal {C}\) and is an idempotent morphism of complexes. We denote by the complex in \(\widetilde{\mathcal {C}}\) with object in degree i and differential .
In the notation of Definition 4.15, the underlying morphism of the differential satisfies
(4.2)
since is a morphism of complexes and consists of idempotents. Furthermore, whenever we write to denote a complex in \(\widetilde{\mathcal {C}}\), we always mean that is an idempotent morphism in \(\textsf {\textrm{Ch}}({\mathcal {C}})\) and that is the induced object in \(\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})\) as described in Definition 4.15.
We make a further remark on the notation . Because of the need to tweak the differentials in according to (4.2), one cannot recover the original complex with differentials from the pair defined in Definition 4.15. This is in contrast to the description of an object in \(\widetilde{\mathcal {C}}\) as a pair (X, e) where one can recover \(X\in \mathcal {C}\) uniquely. Thus, is an abuse of notation but should hopefully cause no confusion.
Lemma 4.14 allows us to induce morphisms of complexes in \(\widetilde{\mathcal {C}}\) given a morphism between complexes in \(\mathcal {C}\) if the complexes involved come with idempotent endomorphisms. The proof is also straightforward.
Lemma 4.16
Suppose that are objects in \(\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})\) and that is a morphism in . Then defining for each \(i\in \mathbb {Z}\) gives rise to a morphism in \(\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})\) with .
Notation 4.17
In the setup of Lemma 4.16, the composite is a morphism of complexes . In this case, we call the underlying morphism of .
We need two more lemmas before we can define a realisation of the functor \(\mathbb {F}\).
Lemma 4.18
Assume . Further, suppose that is an \(\mathfrak {s}\)-distinguished n-exangle and is an idempotent lift of . Then is an n-exangle for \((\widetilde{\mathcal {C}}, \mathbb {F})\).
Proof
Let \((Y,e')\in \widetilde{\mathcal {C}}\) be arbitrary. We will show that the induced sequence
where \((\tilde{d}_i^{(X,e)})_{*} = \widetilde{\mathcal {C}}((Y,e'), \tilde{d}_i^{(X,e)})\), is exact. The exactness of the dual sequence can be verified similarly. Checking the above sequence is a complex is straightforward using that and that .
×
To check exactness at for some \(1\leqslant i \leqslant n\), suppose we have a morphism with \(\tilde{d}_i^{(X,e)} \tilde{r} = 0\), that is, . As , we see that \(d_i^Xr=0\), whence there exists such that \(d_{i-1}^Xs = r\) because is an \(\mathfrak {s}\)-distinguished n-exangle. By Lemma 4.14, there is a morphism with \(t = e_{i-1} s e'\). Then we observe that , whence \(\tilde{d}_{i-1}^{(X,e)} \tilde{t} = \tilde{r}\).
Lastly, suppose is a morphism with . Then we have . Hence, there is a morphism such that as is an \(\mathfrak {s}\)-distinguished n-exangle. Then the morphism with satisfies \(\tilde{d}^{(X,e)}_n \tilde{w} = \tilde{u}\), as required. \(\square \)
Lemma 4.19
Suppose and that in \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\). If and are idempotent lifts of , then are isomorphic in , i.e. .
Proof
We will use [16, Prop. 2.21]. To this end, note that and are both n-exangles in \((\widetilde{\mathcal {C}}, \mathbb {F})\) by Lemma 4.18. Hence, we only have to show that
are both non-empty. Since we have , there are morphisms and in (with and ). We then obtain morphisms and in with and by Lemma 4.16. Note that and . So, since , we have that and \(\tilde{k}_{\bullet }\) are morphisms in and we are done. \(\square \)
Hence, the following is well-defined.
Definition 4.20
For , pick so that and, by Corollary 4.13, an idempotent morphism lifting . We put .
Remark 4.21
For , the definition of depends on neither the choice of with , nor on the choice of lifting by Lemma 4.19. By Corollary 4.13, for each , we can find an \(\mathfrak {s}\)-distinguished n-exangle and an idempotent morphism , such that and is null homotopic in .
Proposition 4.22
The assignment \(\mathfrak {t}\) is an exact realisation of \(\mathbb {F}\).
Proof
(R0) Suppose and , and let \((\tilde{a},\tilde{c}):{\tilde{\delta }} \rightarrow {\tilde{\rho }}\) be a morphism of \(\mathbb {F}\)-extensions. Suppose and . Since (a, c) is a morphism of \(\mathbb {E}\)-extensions, there is a lift of it using that \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\). As and are morphisms in \(\widetilde{\mathcal {C}}\), we have that and . Hence, by Lemma 4.16, it follows that with lifts \((\tilde{a},\tilde{c})\).
(R2) Let \((X,e)\in \widetilde{\mathcal {C}}\) be arbitrary. By Remark 4.5(ii), we have that the zero element of \(\mathbb {F}((0,0),(X,e))\) has the zero element \(_{X}0_{0}\) of \(\mathbb {E}(0,X)\) as its underlying \(\mathbb {E}\)-extension. Since \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\), we know
The tuple is an idempotent morphism lifting . Thus, by Definition 4.20 and using , we see that
Dually, . \(\square \)
×
×
4.3 The Axiom (EA1) for \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\)
Now that we have a biadditive functor and an exact realisation \(\mathfrak {t}\) of \(\mathbb {F}\), we can begin to verify axioms (EA1), (EA2) and (EA2). In this subsection, we will check that the collection of \(\mathfrak {t}\)-inflations is closed under composition. One can dualise the results here to see that \(\mathfrak {t}\)-deflations compose to \(\mathfrak {t}\)-deflations.
The following result only needs that \(\mathfrak {s}\) is an exact realisation of . It is an analogue of [22, Lem. 2.1] for n-exangulated categories, allowing us to complete a “partial” lift of a morphism of extensions.
Lemma 4.23
(Completion Lemma) Let and be \(\mathfrak {s}\)-distinguished n-exangles. Let l, r be integers with \(0 \leqslant l \leqslant r-2 \leqslant n-1\). Suppose there are morphisms and , where , such that is a morphism of \(\mathbb {E}\)-extensions and the solid part of the diagram
commutes. Then there exist morphisms such that (4.3) commutes.
×
Proof
We proceed by induction on \(l \geqslant 0\). Suppose \(l=0\). We induct downwards on \(r \leqslant n+1\). If \(r = n+1\), then the result follows from axiom (R0) for \(\mathfrak {s}\) since is a morphism of \(\mathbb {E}\)-extensions. Now assume that the result holds for \(l=0\) and some \(3 \leqslant r\leqslant n+1\). Suppose we are given morphisms and such that for \(i\in \{ r-1,\ldots , n \}\). By the induction hypothesis, we obtain a morphism
of \(\mathfrak {s}\)-distinguished n-exangles. We will denote this morphism by . Next, note that we have . Since is a weak kernel of , there exists so that . Set for . Notice that we have for \(i \notin \{r-1, r-2\}\). We claim that (4.3) commutes. By construction, we only need to check commutativity of the two squares involving . These indeed commute since
and
using the commutativity of (4.4). This concludes the base case \(l=0\).
×
The inductive step for \(l\geqslant 0\) is carried out in a similar way to the inductive step above on r, using that is a weak cokernel of . \(\square \)
From the Completion Lemma 4.23 and some earlier results from this section we derive the following, which is used in the main result of this subsection.
Lemma 4.24
Suppose is an \(\mathfrak {s}\)-distinguished n-exangle. Assume and are idempotents, such that and
commutes. Then and can be extended to an idempotent morphism with for .
×
Proof
First, suppose \(n = 1\). Then the solid morphisms of the diagram
form a commutative diagram, and by [16, Prop. 3.6(1)] there is a morphism such that is a morphism of \(\mathfrak {s}\)-distinguished 1-exangles. Recall the polynomial from Lemma 4.9. We will show that , where , is the desired idempotent morphism of \(\mathfrak {s}\)-distinguished n-exangles.
×
Since is an \(\mathfrak {s}\)-distinguished 1-exangle, there is an exact sequence
As , there exists a morphism with . This shows that because is idempotent. Hence, is an idempotent morphism of complexes by Lemma 4.9(ii). Furthermore, using Lemma 4.9(i), so that is a morphism of \(\mathbb {E}\)-extensions. This computation also shows the existence of with underlying \(\mathbb {E}\)-extension \(\delta \).
×
Now suppose \(n \geqslant 2\). We have . Therefore, is an element of with underlying \(\mathbb {E}\)-extension \(\delta \). The solid morphisms of the diagram
form a commutative diagram, and is a morphism of \(\mathbb {E}\)-extensions as . Since the rows are the \(\mathfrak {s}\)-distinguished n-exangle , by Lemma 4.23 we can find a morphism , so that the diagram above is a morphism . Furthermore, as and are idempotent, we may assume that is an idempotent by Lemma 4.10. \(\square \)
×
Given a \(\mathfrak {t}\)-inflation \(\tilde{f}\) that fits into a \(\mathfrak {t}\)-distinguished n-exangle , we cannot a priori say too much about how might look. This is one of the main issues in trying to prove (EA1) for \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\). The next lemma gives us a way to deal with this and is the last preparatory result we need before the main result of this subsection.
Lemma 4.25
Let be a \(\mathfrak {t}\)-inflation. Then there is an \(\mathfrak {s}\)-distinguished n-exangle \(\langle X'_{\bullet }, \delta \rangle \) with and for some \(C \in \mathcal {C}\), such that and for some .
Proof
Since is an \(\mathfrak {t}\)-inflation, there is a \(\mathfrak {t}\)-distinguished n-exangle with , and \(\tilde{d}_0^{\widetilde{Y}} = \tilde{f}\). By definition of \(\mathfrak {t}\), this means there is an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y'_{\bullet }, \delta ' \rangle \) with an idempotent morphism \(e'_{\bullet } :\langle Y'_{\bullet }, \delta ' \rangle \rightarrow \langle Y'_{\bullet }, \delta ' \rangle \), such that and , and there are mutually inverse homotopy equivalences which satisfy and . Note that we, thus, have and . In particular, we have a commutative diagram
in \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\), where .
×
Consider the complex and the \(\mathbb {E}\)-extension . Note that if \(n=1\), then ; otherwise we have \({{Y''_{n+1}}} =0\). In either case, we have an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y''_{\bullet }, \delta '' \rangle \) using the axiom (R2) for \(\mathfrak {s}\), and hence also an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y''_{\bullet } \oplus Y'_{\bullet }, \delta '' \oplus \delta ' \rangle \) by [16, Prop. 3.3]. Using the canonical isomorphism we see that the complex
realises in \((\mathcal {C},\mathbb {E},\mathfrak {s})\) by [16, Cor. 2.26(2)]. Consider the diagram
in \(\mathcal {C}\), where and . This diagram commutes since
and
Notice that the composition ba is an automorphism of , and so the complex
forms part of an \(\mathfrak {s}\)-distinguished n-exangle \(\langle X'_{\bullet }, \delta \rangle \) by [16, Cor. 2.26(2)]. We have
as , and . Setting and \(C :=Y'_{1}\) finishes the proof. \(\square \)
×
×
×
We close this subsection with the following result, which together with its dual demonstrates that axiom (EA1) holds for \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\).
Proposition 4.26
Suppose and are \(\mathfrak {t}\)-inflations with . Then is a \(\mathfrak {t}\)-inflation.
Proof
By Lemma 4.25, there exists an \(\mathfrak {s}\)-distinguished n-exangle \(\langle X'_{\bullet }, \delta \rangle \) with and for some \(C \in \mathcal {C}\), so that for some and . Similarly, there is also an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y'_{\bullet }, \delta ' \rangle \) with and for some \(C' \in \mathcal {C}\), so that for some and . Setting , we also have the n-exangle by axiom (R2) for \(\mathfrak {s}\). Then is \(\mathfrak {s}\)-distinguished by [16, Prop. 3.3]. We have and , and
is the \(\mathfrak {s}\)-inflation of with respect to the given decompositions. Since and \(d^{Y'\oplus Y''}_0\) are \(\mathfrak {s}\)-inflations, by (EA1) for \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\), we have that the morphism
is an \(\mathfrak {s}\)-inflation, where we used that . Therefore, there is an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Z''_{\bullet }, \delta '' \rangle \) with , and .
Our next aim is to apply Lemma 4.24 to \(\langle Z''_{\bullet }, \delta '' \rangle \). Thus, we claim that . Since , we can apply [16, Prop. 3.6(1)] to obtain a morphism
of \(\mathfrak {s}\)-distinguished n-exangles. In particular, we have that
(4.5)
As and is idempotent, we see that
(4.6)
This implies that
Since \(\langle X'_{\bullet }, \delta \rangle \) is an \(\mathfrak {s}\)-distinguished n-exangle, by [16, Lem. 3.5] there is an exact sequence
As seen above, vanishes under , so there is a morphism with . Since , this implies
showing that .
×
×
Now consider the idempotent . A quick computation yields the equality . Therefore, by Lemma 4.24, there is an idempotent morphism \(e''_{\bullet } :\langle Z''_{\bullet }, \delta '' \rangle \rightarrow \langle Z''_{\bullet }, \delta '' \rangle \) with , \(e''_{1} = e'_{1} \oplus 0 \oplus 0\) as well as an \(\mathbb {F}\)-extension with underlying \(\mathbb {E}\)-extension \(\rho = \delta ''\). We obtain a \(\mathfrak {t}\)-distinguished n-exangle \(\langle (Z''_{\bullet }, e''_{\bullet }), {\tilde{\rho }} \rangle \). Then the \(\mathfrak {t}\)-inflation of this n-exangle is given by the morphism satisfying
As is an isomorphism in \(\widetilde{\mathcal {C}}\), the complex
with \(\mathfrak {t}\)-inflation and forms part of the \(\mathfrak {t}\)-distinguished n-exangle \(\langle \widetilde{X}''_{\bullet }, {\tilde{\rho }} \rangle \) by [16, Cor. 2.26(2)]. \(\square \)
×
4.4 The Axiom (EA2) for \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\)
The goal of this subsection is to show that axiom (EA2) holds for the triplet \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\). Again, by dualising one can deduce that axiom (EA2) also holds. We need two key technical lemmas first.
Lemma 4.27
Suppose that:
(i)
is an \(\mathbb {F}\)-extension;
(ii)
is a morphism in \(\widetilde{\mathcal {C}}\) for some ;
(iii)
and are \(\mathfrak {s}\)-distinguished n-exangles with ;
(iv)
is an idempotent morphism lifting , such that is null homotopic; and
(v)
is an idempotent morphism lifting , such that is null homotopic.
Then a good lift of the morphism of \(\mathbb {E}\)-extensions exists, so that
is commutative in \(\mathcal {C}\). In particular, we have as morphisms .
×
Remark 4.28
Notice that and \( c e'_{n+1}= c\) imply
(4.8)
Therefore, is indeed a morphism of \(\mathbb {E}\)-extensions and condition (v) makes sense. Condition (iv) makes sense due to Remark 4.5(iv).
Proof of Lemma 4.27
Since is a morphism of \(\mathbb {E}\)-extensions, it admits a good lift using axiom (EA2) for the n-exangulated category \((\mathcal {C},\mathbb {E},\mathfrak {s})\). Define for \(0\leqslant i \leqslant n+1\). Note that and by assumption. For \(i=0\), we have . On the other hand, for \(i=n+1\) we have . Therefore, the morphism is of the form .
The squares on the top and bottom faces in (4.7) commute as and , respectively, are morphisms of complexes. The squares on the front and back faces in (4.7) commute because is the sum of morphisms of complexes from to . This also implies is a lift of . Of the remaining squares, the leftmost clearly commutes and the rightmost commutes as is a morphism in \(\widetilde{\mathcal {C}}\). For \(1\leqslant i \leqslant n\), we have
Therefore, diagram (4.7) commutes and, further, the last assertion follows.
It remains to show that is a good lift of . Recall that and are both null homotopic by assumption, and so is also null homotopic. Then it follows from [16, Rem. 2.33(1)] that is a good lift of since \(g'_{\bullet }\) is. \(\square \)
The next result allows us to define a good lift in \(\widetilde{\mathcal {C}}\) from the one we created in Lemma 4.27.
Lemma 4.29
In the setup of Lemma 4.27, the morphism with under-lying morphism is a good lift of the morphism of \(\mathbb {F}\)-extensions.
Proof
From (4.8), we see that is indeed an \(\mathbb {F}\)-extension and a morphism of \(\mathbb {F}\)-extensions. Using and , as well as the commutativity of (4.7), we see that is a morphism of \(\mathfrak {t}\)-distinguished n-exangles, lifting .
Recall from Definition 3.4 that denotes the mapping cone of in \(\mathcal {C}\), and that is \(\mathfrak {s}\)-distinguished as is a good lift of . Using the commutativity of (4.7), that and are morphisms of complexes, and that , one can verify that the diagram
commutes. Thus, the vertical morphisms form an idempotent morphism \({e''_{\bullet } :{M_{g}^\mathcal {C}}_{\bullet } \rightarrow {M_{g}^\mathcal {C}}_{\bullet }}\) of complexes. Furthermore, (4.9) is a morphism of \(\mathfrak {s}\)-distinguished n-exangles as
This calculation also shows that . Thus, by definition of \(\mathfrak {t}\), we have that \(\mathfrak {t}({\tilde{\rho }}) = [ ({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet }) ]\), i.e. \(\langle ({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet }), {\tilde{\rho }} \rangle \) is \(\mathfrak {t}\)-distinguished.
×
It is straightforward to verify that the object \(({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet })\) is equal to the mapping cone \({M^{\widetilde{\mathcal {C}}}_{\tilde{h}}}_{\bullet }\) of in , so \(\langle {M^{\widetilde{\mathcal {C}}}_{\tilde{h}}}_{\bullet }, {\tilde{\rho }} \rangle \) is \(\mathfrak {t}\)-distinguished. Lastly, we note that because
Hence, is a \(\mathfrak {t}\)-distinguished n-exangle. \(\square \)
We are in position to prove axiom (EA2) for \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\). Axiom (EA2) can be shown dually.
Proposition 4.30
(Axiom (EA2) for \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\)) Let -extension and suppose is a morphism in \(\widetilde{\mathcal {C}}\). Suppose are \(\mathfrak {t}\)-distinguished n-exangles. Then has a good lift .
Proof
Notice that the underlying \(\mathbb {E}\)-extension of is . By definition of \(\mathfrak {t}\) and Remark 4.21, there are \(\mathfrak {s}\)-distinguished n-exangles \(\langle X'_{\bullet }, \delta \rangle \) and and idempotent morphisms and , such that and \(\mathfrak {t}(\tilde{c}^\mathbb {F}{\tilde{\delta }}) = [(Y'_{\bullet }, e'_{\bullet })]\), and so that and are null homotopic in . We note that since and , we have that and, in particular, that . Moreover, it follows that all the hypotheses of Lemma 4.27 are satisfied.
Therefore, by Lemma 4.29, the morphism of \(\mathbb {F}\)-extensions has a good lift . Since and , there is a homotopy equivalence in and a homotopy equivalence in . By [16, Cor. 2.31], the composite is then also a good lift of . \(\square \)
4.5 Main Results
In this subsection we present our main results regarding the idempotent completion and an n-exangulated structure we can impose on it.
Definition 4.31
We call an n-exangulated category \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) (resp. weakly) idempotent complete if the underlying additive category \(\mathcal {C}\) is (resp. weakly) idempotent complete.
In [5, Prop. 2.5] a characterisation of weakly idempotent complete extriangulated categories is given. Next we note that the first part of Theorem A from Sect. 1 summarises our work from Sects. 4.1–4.4.
Theorem 4.32
Let \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) be an n-exangulated category. Then the triplet \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is an idempotent complete n-exangulated category.
Proof
This follows from Propositions 2.5, 4.22, 4.26 and 4.30, and the duals of the latter two. \(\square \)
And Corollary C from Sect. 1 is a nice consequence of this.
Corollary 4.33
Let \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) be an n-exangulated category, such that each object in \(\mathcal {C}\) has a semi-perfect endomorphism ring. Then the idempotent completion \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is a Krull-Schmidt n-exangulated category.
Proof
By Theorem 4.32, the idempotent completion \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is an idempotent complete n-exangulated category. By [12, Thm. A.1] (or [24, Cor. 4.4]), it is enough to show that endomorphism rings of objects in \(\widetilde{\mathcal {C}}\) are semi-perfect rings. Let (X, e) be an object in \(\widetilde{\mathcal {C}}\). We have that is semi-perfect since is fully faithful (see Proposition 2.5). By Remark 2.6, we have that . In particular, we see that is an idempotent subring of the semi-perfect ring . Hence, by Anderson–Fuller [1, Cor. 27.7], we have that the endomorphism ring of each object in \(\widetilde{\mathcal {C}}\) is semi-perfect. \(\square \)
We recall that, by [23, Cor. 4.12], an n-exangulated category is n-exact if and only if its inflations are monomorphisms and its deflations are epimorphisms.
Corollary 4.34
Suppose \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) is n-exact. Then \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is n-exact.
Proof
We use [23, Cor. 4.12] and only show that \(\mathfrak {t}\)-inflations are monomorphisms; showing \(\mathfrak {t}\)-deflations are epimorphisms is dual. Let be a \(\mathfrak {t}\)-inflation and suppose there is a morphism in \(\widetilde{\mathcal {C}}\) with \(\tilde{f} \tilde{g} = \widetilde{0}\). By Lemma 4.25, there is an \(\mathfrak {s}\)-inflation , which is monic as \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) is n-exact. We have \(fg = 0\) as \(\tilde{f} \tilde{g} = \widetilde{0}\), and we also have because the underlying morphism g of \(\tilde{g}\) satisfies . Thus, we see that and this implies \(g = 0\) as is monic. Hence, \(\tilde{g} = 0\) and we are done. \(\square \)
The main aim of this subsection is to establish the relevant 2-universal property of the inclusion functor . We will show that forms part of an n-exangulated functor , and that this is 2-universal in an appropriate sense. The next lemma is straightforward to check.
Lemma 4.35
The family of abelian group homomorphisms
for , defines a natural isomorphism .
Proposition 4.36
The pair is an n-exangulated functor from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\).
Proof
We verify that if is an \(\mathfrak {s}\)-distinguished n-exangle, then is \(\mathfrak {t}\)-distinguished, where . We have the idempotent morphism , so from Definition 4.20 we see that . \(\square \)
We lay out some notation that will be used in the remainder of this section and also in Sect. 5.
Notation 4.37
Let (X, e) be an object in the idempotent completion \(\widetilde{\mathcal {C}}\) of \(\mathcal {C}\). Then (X, e) is a direct summand of by Remark 2.6. By and , we denote the canonical inclusion and projection morphisms, respectively.
Recall that, for an additive category \(\mathcal {C}'\) and a biadditive functor , the \(\mathbb {E}'\)-attached complexes and morphisms between them were defined in Definition 3.1, and together they form an additive category.
Lemma 4.38
Let be an \(\mathbb {F}\)-extension. Suppose and is an idempotent morphism. With , we have that
(4.10)
as \(\mathfrak {t}\)-distinguished n-exangles.
Proof
Let . First, note that and are \(\mathbb {F}\)-attached complexes, and is a \(\mathfrak {t}\)-distinguished n-exangle since \((\mathscr {F},\Gamma )\) is an n-exangulated functor.
Consider the morphisms and of complexes induced by , as well as the corresponding ones \(\tilde{i}_{e'_{\bullet }}\) and \(\tilde{p}_{e'_{\bullet }}\) for \(e'_{\bullet }\). We claim that there is a biproduct diagram
in the category of \(\mathbb {F}\)-attached complexes. To see that is a morphism of \(\mathbb {F}\)-attached complexes, we just need to check that . By Remark 4.7, it is enough to see that holds, and this is indeed true because is a morphism of \(\mathbb {F}\)-attached complexes. To see that \(\tilde{i}_{e'_{\bullet }}\) and \(\tilde{p}_{e'_{\bullet }}\) are morphisms of \(\mathbb {F}\)-attached complexes, one uses that . Furthermore, we have the identities , and , so (4.11) is a biproduct diagram in the additive category of \(\mathbb {F}\)-attached complexes. Therefore, we have that (4.10) is an isomorphism as \(\mathbb {F}\)-attached complexes.
×
Lastly, since is \(\mathfrak {t}\)-distinguished, it follows from [16, Prop. 3.3] that (4.10) is an isomorphism of \(\mathfrak {t}\)-distinguished n-exangles. \(\square \)
Thus, we can now present and prove the main result of this section, which shows that the n-exangulated inclusion functor is 2-universal amongst n-exangulated functors from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to idempotent complete n-exangulated categories.
Theorem 4.39
Suppose \((\mathscr {F},\Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) is an n-exangulated functor to an idempotent complete n-exangulated category \((\mathcal {C}', \mathbb {E}', \mathfrak {s}')\). Then the following statements hold.
(i)
There is an n-exangulated functor \((\mathscr {E}, \Psi ) :(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) and an n-exangulated natural isomorphism .
(ii)
In addition, for any n-exangulated functor \((\mathscr {G}, \Theta ) :(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) and any n-exangulated natural transformation , there is a unique n-exangulated natural transformation with .
Proof
(i) By Proposition 2.8(i), there exists an additive functor \(\mathscr {E}:\widetilde{\mathcal {C}} \rightarrow \mathcal {C}'\) and a natural isomorphism . It remains to show that \(\mathscr {E}\) forms part of an n-exangulated functor \((\mathscr {E},\Psi )\) and that is n-exangulated.
First, we define a natural transformation \(\Psi :\mathbb {F}(-,-) \Rightarrow \mathbb {E}'(\mathscr {E}-, \mathscr {E}-)\) as the composition of several abelian group homomorphisms. For , we set
For and in \(\widetilde{\mathcal {C}}\), we define an abelian group homomorphism
and put
For morphisms and in \(\mathcal {C}\), we have
(4.12)
as is natural. For morphisms , we have
(4.13)
using how \(\mathbb {F}\) is defined on morphisms (see Definition 4.4). We claim that the family of abelian group homomorphisms
for defines a natural transformation \(\Psi :\mathbb {F}(-,-) \Rightarrow \mathbb {E}'(\mathscr {E}-, \mathscr {E}-)\). To this end, fix objects and , and morphisms and in \(\widetilde{\mathcal {C}}\). First, note that we have
(4.14)
and, similarly,
(4.15)
Therefore, we see that
Next, we must show that \((\mathscr {E},\Psi )\) sends \(\mathfrak {t}\)-distinguished n-exangles to \(\mathfrak {s}'\)-distinguished n-exangles. Thus, let and , and suppose . We need that , which will follow from seeing that is a direct summand of an \(\mathfrak {s}'\)-distinguished n-exangle.
By Remark 4.21, we may take a complex in with and an idempotent morphism lifting , such that . Note for later that we thus have , and hence . Let . Since \((\mathscr {F}, \Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) is an n-exangulated functor, the n-exangle is \(\mathfrak {s}'\)-distinguished. As we have an isomorphism of complexes , the \(\mathbb {E}'\)-attached complex is \(\mathfrak {s}'\)-distinguished by [16, Cor. 2.26(2)]. Since is just the identity homomorphism, a quick computation yields
(4.16)
In particular, this implies that is \(\mathfrak {s}'\)-distinguished.
Note that as \(\mathbb {F}\)-attached complexes by Lemma 4.38, where . We see that is a direct summand of the \(\mathfrak {s}'\)-distinguished n-exangle by Proposition 3.7(ii). Hence, by [16, Prop. 3.3], and so \((\mathscr {E},\Psi )\) is an n-exangulated functor.
Lastly, it follows immediately from (4.16) that is an n-exangulated natural transformation .
(ii) By Proposition 2.8(ii), there exists a unique natural transformation with , so it remains to show that induces an n-exangulated natural transformation \((\mathscr {E}, \Psi ) \Rightarrow (\mathscr {G}, \Theta )\). For this, let and be arbitrary. Note that we have
(4.17)
Hence, we obtain
and the proof is complete. \(\square \)
We close this section with some remarks on our main results and constructions.
Remark 4.40
Before commenting on how our results unify the constructions in cases in the literature and on how our proof methods compare, we set up and recall a little terminology. Suppose \((\mathcal {C},\mathbb {E},\mathfrak {s})\) and \((\mathcal {C}',\mathbb {E}',\mathfrak {s}')\) are n-exangulated categories. We call an n-exangulated functor \((\mathscr {F},\Gamma ) :(\mathcal {C},\mathbb {E},\mathfrak {s}) \rightarrow (\mathcal {C}',\mathbb {E}',\mathfrak {s}')\) an n-exangulated isomorphism if \(\mathscr {F}\) is an isomorphism of categories and \(\Gamma \) is a natural isomorphism. This terminology is justified by [9, Prop. 4.11]. Lastly, we recall that n-exangulated functors between \((n+2)\)-angulated categories are \((n+2)\)-angulated in the sense of [10, Def. 2.7] (or exact as in Bergh–Thaule [8, Sec. 4]), and that n-exangulated functors between n-exact categories are n-exact in the sense of [10, Def. 2.18]; see [10, Thms. 2.33, 2.34].
It has been shown that a triplet \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is a 1-exangulated category if and only if it is extriangulated (see [16, Prop. 4.3]). Suppose that \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an extriangulated category and consider the idempotent completion \(\widetilde{\mathcal {C}}\) of \(\mathcal {C}\). By [27, Thm. 3.1], there is an extriangulated structure \((\mathbb {F}',\mathfrak {t}')\) on \(\widetilde{\mathcal {C}}\). By our Theorem 4.32, there is a 1-exangulated (or extriangulated) category \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\). By direct comparison of the constructions, one can check that \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) and \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\) are n-exangulated isomorphic. Indeed, the bifunctors \(\mathbb {F}\) and \(\mathbb {F}'\) differ only by a labelling of the elements due to our convention in Definition 4.4; and, ignoring this re-labelling, the realisations \(\mathfrak {s}\) and \(\mathfrak {s}'\) are the same by Lemma 4.19. Furthermore, since \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\) recovers the triangulated and exact category cases, we see that our construction agrees with the classical (i.e. \(n=1\)) cases up to n-exangulated isomorphism.
For larger n, we just need to compare \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) with the construction in [25]. Thus, suppose \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is the n-exangulated category coming from an \((n+2)\)-angulated category . Recall that in this case \(\mathbb {E}(Z,X) = \mathcal {C}(Z,\Sigma X)\) for \(X,Z\in \mathcal {C}\). Using [25, Thm. 3.1], one obtains an \((n+2)\)-angulated category , where \(\widetilde{\Sigma }\) is induced by \(\Sigma \). From this \((n+2)\)-angulated category, just like above, we obtain an induced n-exangulated category \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\). Notice that \(\mathbb {F}'(-,-) = \widetilde{\mathcal {C}}(-,\widetilde{\Sigma }-)\). Comparing \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\) to the n-exangulated category \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) found from Theorem 4.32, again we see that \(\mathbb {F}\) and \(\mathbb {F}'\) differ by the labelling convention we chose in Definition 4.4. By [16, Prop. 4.8] we have that \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) induces an \((n+2)\)-angulated category , and therefore the n-exangulated inclusion functor is, moreover, \((n+2)\)-angulated. It follows from [25, Thm. 3.1(2)] that and must be equal, and hence \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) and \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\) are n-exangulated isomorphic.
Remark 4.41
Our proofs in this article differ from the proofs in both the extriangulated and the \((n+2)\)-angulated cases. First, the axioms for an n-exangulated category look very different from the axioms for an extriangulated category. Therefore, the proofs from [27] cannot be directly generalised to the \(n>1\) case. Even of the results that seem like they might generalise nicely, one comes across immediate obstacles. Indeed, Lin [25, p. 1064] already points out that lifting idempotent morphisms of extensions to idempotent morphisms of n-exangles is non-trivial. Despite this, we are able to overcome this here. This, amongst other problems, forces Lin to use another approach, and hence demonstrates why our methods are distinct.
Remark 4.42
He–He–Zhou [15] have considered idempotent completions of n-exangulated categories in a specific setup. In their setup, there is an ambient Krull-Schmidt \((n+2)\)-angulated category \(\mathcal {C}\) and an additive subcategory \(\mathcal {A}\) that is n-extension-closed (see Definition 5.2) and closed under direct summands in \(\mathcal {C}\). The main aim of [15] is to show that the idempotent completion \(\widetilde{\mathcal {A}}\) of \(\mathcal {A}\) is an n-exangulated subcategory of \(\widetilde{\mathcal {C}}\).
Since \(\mathcal {A}\) is an additive subcategory of and closed under direct summands in a Krull-Schmidt category, it is Krull-Schmidt itself. In particular, \(\mathcal {A}\simeq \widetilde{\mathcal {A}}\) is already idempotent complete by [24, Cor. 4.4]. Moreover, in the setup of [15], it already follows that \(\mathcal {A}\) inherits an n-exangulated structure from \(\mathcal {C}\simeq \widetilde{\mathcal {C}}\). Indeed, (EA1) is proven in [22, Lem. 3.8], and (EA2) and (EA2) are straightforward to check directly. It is then clear that \(\mathcal {A}\) inherits an n-exangulated structure from \(\mathcal {C}\).
5 The Weak Idempotent Completion of an n-Exangulated Category
Just as in Sect. 4, we assume \(n\geqslant 1\) is an integer and that \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an n-exangulated category. By Theorems 4.32 and 4.39, the idempotent completion of \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an n-exangulated category \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) and the inclusion functor of \(\mathcal {C}\) into \(\widetilde{\mathcal {C}}\) is part of an n-exangulated functor , which satisfies the 2-universal property from Theorem 4.39. In this section, we turn our attention to the weak idempotent completion \(\widehat{\mathcal {C}}\) of \(\mathcal {C}\) and we show that it forms part of a triplet \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\) that is n-extension-closed (see Definition 5.2) in \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\). It will then follow that \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\) is itself n-exangulated, and, moreover, there is an analogue of Theorem 4.39 for \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\); see Theorem 5.5.
We begin with the following proposition, which is an analogue of Lemma 4.38 for the weak idempotent completion.
Proposition 5.1
Suppose are objects, is an \(\mathbb {F}\)-extension and . Then there is a \(\mathfrak {t}\)-distinguished n-exangle with and an \(\mathfrak {s}\)-distinguished n-exangle , such that
as \(\mathfrak {t}\)-distinguished n-exangles.
Proof
By Corollary 4.13, there exists an idempotent morphism with for \(2 \leqslant i \leqslant n-1\), as well as a homotopy , where . Notice for \(i = 0, n+1\) by assumption. Furthermore, for . By Lemma 4.38 we have -distinguished n-exangles. We will show that there is an isomorphism in for some , as well as an isomorphism in for some object .
If \(i = 0,n+1\), then is split by assumption, so by Lemma 2.4 there are objects \(Y'_i \in \mathcal {C}\) and isomorphisms . For \(2\leqslant i \leqslant n-1\), we see that , so by Lemma 2.4 again we have isomorphisms , but now where \( Y'_{i} = 0 \in \mathcal {C}\). Since by assumption and because for \(2 \leqslant i \leqslant n-1\), we put and for \(i \in \{ 0,n+1\} \cup \{2,\ldots , n-1\}\). It remains to find appropriate isomorphisms and \(\tilde{s}'_i\) for \(i=1,n\).
We have a morphism with underlying morphism and another with underlying morphism by Lemma 4.14. Since is a homotopy, we see that . This implies
(5.1)
and so . Similarly, we also have .
1.
If \(n=1\), then (5.1) shows that is a section in the complex , and hence this complex is a split short exact sequence by [16, Claim 2.15]. In particular, we have that . So we put \(Y'_{1} :=Y'_{0}\oplus Y'_{2}\) and define to be this composition of isomorphisms. As , and and are isomorphic to objects in \(\widehat{\mathcal {C}}\), by Lemma 2.12 there is an isomorphism for some .
2.
If \(n \geqslant 2\), then the form of the homotopy implies that the identities and hold. Therefore, we see that
which shows that and are mutually inverse isomorphisms. We now define \(Y'_{1} :=Y'_{0}\) and . Because there are isomorphisms , and and are isomorphic to objects in \(\widehat{\mathcal {C}}\), by Lemma 2.12 there is an isomorphism for some . In a similar way, one can show that and are mutually inverse isomorphisms. We set \(Y'_{\textit{n}} :=Y'_{n+1}\) and . In addition, there is an isomorphism for some .
The complex with object in degree \(0\leqslant i \leqslant n+1\) and differential in degree \(0\leqslant i\leqslant n\) is isomorphic to via . Furthermore, as and are identity morphisms, we have that is \(\mathfrak {t}\)-distinguished by [16, Cor. 2.26(2)]. The complex \(\widetilde{Y}'_{\bullet }\) with object in degree \(0\leqslant i \leqslant n+1\) and differential in degree \(0\leqslant i \leqslant n\) is isomorphic to via \(\tilde{s}'_{\bullet }\). Moreover, this induces an isomorphism of \(\mathbb {F}\)-attached complexes, and hence of \(\mathfrak {t}\)-distinguished n-exangles. It is clear that
by the construction of \(\widetilde{Y}'_\bullet \). Lastly, is \(\mathfrak {s}\)-distinguished using [16, Prop. 3.3] and that \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\). \(\square \)
From Proposition 5.1 we see that \(\widehat{\mathcal {C}}\) is n-extension-closed in \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) in the following sense.
Definition 5.2
[17, Def. 4.1] Let \((\mathcal {C}',\mathbb {E}',\mathfrak {s}')\) be an n-exangulated category. A full subcategory \(\mathcal {D}\subseteq \mathcal {C}'\) is said to be n-extension-closed if, for all \(A,C\in \mathcal {D}\) and each \(\mathbb {E}'\)-extension \(\delta \in \mathbb {E}'(C,A)\), there is an object such that for all \(1\leqslant i \leqslant n\) and .
Let us now define the biadditive functor and realisation with which we wish to equip \(\widehat{\mathcal {C}}\).
Definition 5.3
(i)
Let be the restriction of .
(ii)
For a \(\mathbb {G}\)-extension , there is a \(\mathfrak {t}\)-distinguished n-exangle with by Proposition 5.1. We put , the isomorphism class of in .
(iii)
Recall from Sect. 2.2 that is the inclusion functor defined by on objects \(X \in \mathcal {C}\). Let be the restriction of the natural transformation defined in Lemma 4.35. This means for .
Since \(\widehat{\mathcal {C}}\) is an n-extension closed subcategory of \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) by Proposition 5.1, one can use [17, Prop. 4.2(1)] to deduce axioms (EA2) and (EA2) hold for the triplet \((\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})\). The difficult part is then to show that (EA1) is satisfied; this follows from Lemma 5.4 below. We note here, however, it has been shown in [23, Thm. A] that any n-extension-closed subcategory of an n-exangulated category that is also closed under isomorphisms inherits an n-exangulated structure in the expected way. Although the isomorphism-closure in [23] is assumed only for convenience, we highlight that the weak idempotent completion is not necessarily closed under isomorphisms in the idempotent completion using the constructions in Sect. 2. Indeed, one can show that \(\widehat{\mathcal {C}}\) is isomorphism-closed in \(\widetilde{\mathcal {C}}\) if and only if \(\mathcal {C}\) is already weakly idempotent complete.
Lemma 5.4
Let be a \(\mathfrak {t}\)-inflation with . Then there is a \(\mathfrak {t}\)-distinguished n-exangle with and .
Proof
By Lemma 4.25 there is an object \(C \in \mathcal {C}\), a morphism and an \(\mathfrak {s}\)-distinguished n-exangle and . The solid morphisms of the diagram
form a commutative diagram. By Lemma 4.24 there exists an idempotent morphism of n-exangles , and for \(3 \leqslant i \leqslant n+1\), which makes the diagram above commute. Let and . Notice that the underlying \(\mathbb {E}\)-extension of \({\tilde{\rho }}'\) is \(\rho \). Set . Then as \(\mathfrak {t}\)-distinguished n-exangles by Lemma 4.38.
×
We claim that there is a split short exact sequence
Since realises the trivial \(\mathbb {F}\)-extension , we have that is a section by [16, Claim 2.15]. If \(n=1\), then this is enough to see that (5.2) is split short exact. For \(n\geqslant 2\) we notice that there is an isomorphism in \(\widetilde{\mathcal {C}}\), so . Thus, since is a weak kernel of , we see that factors through . In particular, this implies is a cokernel of . Again, (5.2) is split short exact.
×
In particular, we have an isomorphism . We know that the objects and are isomorphic to objects in by Lemma 2.4, as and are split idempotents. This implies that is isomorphic to an object in \(\widehat{\mathcal {C}}\) by Lemma 2.12. Again Lemma 2.12 and the isomorphism imply that there is an isomorphism for some .
The morphism with underlying morphism is an isomorphism. Finally, put for \(i = 0\) and \(3 \leqslant i \leqslant n+1\). Then the complex
is isomorphic to via in . With , we see that is \(\mathfrak {t}\)-distinguished by [16, Cor. 2.26(2)], as desired. \(\square \)
×
We may state and prove our main result of this section.
Theorem 5.5
Suppose that \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) is an n-exangulated category. Then \((\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})\) is a weakly idempotent complete n-exangulated category, and is an n-exangulated functor, such that the following 2-universal property is satisfied. Suppose \((\mathscr {F},\Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) is an n-exangulated functor to a weakly idempotent complete n-exangulated category \((\mathcal {C}', \mathbb {E}', \mathfrak {s}')\). Then the following statements hold.
(i)
There is an n-exangulated functor \((\mathscr {E}, \Psi ) :(\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) and an n-exangulated natural isomorphism .
(ii)
In addition, for any n-exangulated functor \((\mathscr {G}, \Theta ) :(\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) and any n-exangulated natural transformation , there is a unique n-exangulated natural transformation with .
Proof
Since \(\widehat{\mathcal {C}}\) is a full subcategory of \(\widetilde{\mathcal {C}}\) and because \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\) is n-exangulated, we can apply [17, Prop. 4.2] and Definition 5.2. We showed above that \(\widehat{\mathcal {C}}\) is n-extension-closed in \(\widetilde{\mathcal {C}}\); see Proposition 5.1. Moreover, it follows immediately from Lemma 5.4 and its dual that \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\) satisfies (EA1). Therefore, we deduce that \((\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})\) is an n-exangulated category.
One may argue that is an n-exangulated functor as in Proposition 4.36, by using the definition of \(\mathfrak {r}\) and noting that lies in \(\widehat{\mathcal {C}}\) for all \(X\in \mathcal {C}\).
(i) One argues like in the proof of Theorem 4.39(i), but using Proposition 2.13 instead of Proposition 2.8, and Proposition 5.1 instead of Lemma 4.38. In particular, we note that the isomorphism of \(\mathfrak {t}\)-distinguished n-exangles from the statement of Proposition 5.1 induces an isomorphism
of \(\mathfrak {r}\)-distinguished n-exangles.
(ii) Similarly, one adapts the proof of Theorem 4.39(ii), using Proposition 2.13 instead of Proposition 2.8. \(\square \)
Finally, we have an analogue of Corollary 4.34 as a consequence.
Corollary 5.6
Suppose \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) is n-exact. Then \((\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})\) is n-exact.
Proof
The n-exangulated category \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) is n-exact by Corollary 4.34. As \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\) inherits its structure as an n-extension closed subcategory of \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\), the result follows from the proof of [23, Cor. 4.15]. \(\square \)
Acknowledgements
The authors would like to thank Theo Bühler, Ruben Henrard and Adam-Christiaan van Roosmalen for useful email communications, and Andrew Brooke-Taylor and Peter Jørgensen for helpful discussions. The authors are very grateful to the referees for their comments and suggestions. In particular, these comments led to Corollaries 4.34 and 5.6, and improvements to the exposition.
Declarations
Conflicts of interest
Not applicable.
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given by
on objects. Moreover, in several of the cases above it has been shown that this functor is 2-universal in an appropriate sense; see e.g. Proposition 2.8 for a precise formulation. For example, without any assumptions other than additivity, the functor
is additive and 2-universal amongst additive functors from \(\mathcal {C}\) to idempotent complete additive categories. On the other hand, if e.g. \(\mathcal {C}\) has an exact structure, then
is exact and 2-universal amongst exact functors from \(\mathcal {C}\) to idempotent complete exact categories.
is a biadditive functor to the category of abelian groups, and \(\mathfrak {s}\) is a so-called additive realisation of \(\mathbb {E}\). The realisation \(\mathfrak {s}\) associates to each \(\delta \in \mathbb {E}(Z,X)\) a certain equivalence class
of a 3-term complex. As an example, each triangulated category \((\mathcal {C},\Sigma ,{{\triangle }})\), where \(\Sigma \) is a suspension functor and \({{\triangle }}\) is a triangulation, is an extriangulated category. Indeed, one defines the corresponding bifunctor by
. See [30, Prop. 3.22] for more details. In addition, each suitable exact category is extriangulated; see [30, Exam. 2.13]. A particular advantage of this theory is that the collection of extriangulated categories is closed under taking extension-closed subcategories. Although an extension-closed subcategory of an exact category is again exact, the same does not hold in general for triangulated categories.
satisfying some axioms, where \(\Sigma \) is still an automorphism of \(\mathcal {C}\), but now
consists of a collection of \((n+2)\)-angles each of which has \(n+3\) terms.
, and a so-called exact realisation \(\mathfrak {s}\) of \(\mathbb {E}\), which satisfy some axioms (see Sect. 3.1). The realisation \(\mathfrak {s}\) now associates to each
a certain equivalence class (see Sect. 3.1) 
is called an \(\mathfrak {s}\)-distinguished n-exangle. We recall that structure-preserving functors between n-exangulated categories were defined in [10, Def. 2.32]. They are known as n-exangulated functors and they send distinguished n-exangles to distinguished n-exangles.
as follows. For any pair of objects \((X,e),(Z,e')\in \widetilde{\mathcal {C}}\), we let \(\mathbb {F}((Z,e'),(X,e))\) consist of triplets \((e,\delta ,e')\) where \(\delta \in \mathbb {E}(Z,X)\) such that \(\mathbb {E}(Z,e)(\delta ) = \delta = \mathbb {E}(e',X)(\delta )\). On morphisms \(\mathbb {F}\) is essentially a restriction of \(\mathbb {E}\); see Definition 4.4 for details. Now we define a realisation \(\mathfrak {t}\) of \(\mathbb {F}\). For \((e,\delta ,e') \in \mathbb {F}((Z,e'),(X,e))\), we have that
for some \((n+2)\)-term complex
with
and
since \(\mathfrak {s}\) is a realisation of \(\mathbb {E}\). We choose an idempotent morphism
of complexes, such that
and
; see Corollary 4.13. Lastly, we set \(\mathfrak {t}((e,\delta ,e'))\) to be the equivalence class of the complex 
extends to an n-exangulated functor
, which is 2-universal among n-exangulated functors from \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) to idempotent complete n-exangulated categories.
of n-extensions of C by A forms a set; see [16, Prop. 4.34]. As in [23, Def. 4.6], we say that an n-exangulated category \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) is n-exact if its n-exangulated structure arises in this way. Combining Theorem A with [23, Cor. 4.12], we deduce the following.
. The category \(\mathcal {A}\) is idempotent complete, or has split idempotents, if every idempotent in \(\mathcal {A}\) splits.
(see e.g. Auslander [3, p. 188]). In particular, the idempotent e and its counterpart
each admit a kernel. Idempotent complete additive categories can be characterised by such a criterion and its dual.
is idempotent. For objects \((X,e), (Y,e') \in \hbox {obj}\widetilde{\mathcal {A}}\), a morphism from (X, e) to \((Y,e')\) is a triplet \((e',r,e)\), where \(r \in \mathcal {A}(X,Y)\) satisfiesin \(\mathcal {A}\). Composition of morphisms is defined bywhenever \((e',r,e)\in \widetilde{\mathcal {A}}((X,e), (Y, e'))\) and \((e'',s,e')\in \widetilde{\mathcal {A}}((Y,e'), (Z, e''))\). The identity of an object \((X,e)\in \hbox {obj}\widetilde{\mathcal {A}}\) will be denoted
and is the morphism (e, e, e).
is defined as follows. An object \(X \in \hbox {obj}\mathcal {A}\) is sent to
and a morphism \(r \in \mathcal {A}(X,Y)\) is mapped to
.
is a split idempotent, with a splitting \(e=sr\) where \(r :X \rightarrow Y\) and \(s :Y \rightarrow X\), then
.
and
. Hence, there are morphisms
and
in \(\widetilde{\mathcal {A}}\) with
and
. Hence, \(\tilde{r}\) and \(\tilde{s}\) are mutually inverse isomorphisms in \(\widetilde{\mathcal {A}}\). \(\square \)
is an equivalence of categories; see e.g. [11, Rem. 6.5]. But more generally we have the following.
is fully faithful and additive.
. Indeed, there is an isomorphism
. The canonical inclusion of (X, e) into
is given by the morphism
, and the projection of
onto (X, e) by
. Similarly for
.
is 2-universal in some sense; see Proposition 2.8. For this we recall the notion of whiskering a natural transformation by a functor. We will use Hebrew letters (e.g.
(beth),
(tsadi),
(daleth),
(mem)) for natural transformations. Suppose \(\mathcal {B},\mathcal {C},\mathcal {D}\) are categories and that we have a diagram 
is a natural transformation.
is the natural transformation
defined by
for each \(X\in \mathcal {B}\).
.
and, in addition,
, there exists a unique natural transformation
with
.
also has kernel k. Conversely, if \(e:X\rightarrow X\) is a split idempotent, with splitting given by \(e=sr\) where \(r :X \rightarrow Y\), then a kernel of e is also a kernel of r. Therefore, weakly idempotent complete categories are those additive categories in which split idempotents admit kernels, in contrast to idempotent complete categories in which all idempotents admit kernels (see Proposition 2.2).
is a split idempotent in \(\mathcal {A}\).
, where \(r:X\rightarrow Y\) and \(s:Y\rightarrow X\), then
in \(\widetilde{\mathcal {A}}\) by Lemma 2.4. That is, we have not added any objects that are not already isomorphic to some object of
. On the other hand, if we take objects in \(\widehat{\mathcal {A}}\) to be pairs (X, e) where
splits (as in Definition 2.10), then we have
in \(\widehat{\mathcal {A}}\), where
and
, where \(r':X\rightarrow Y'\) and \(s':Y'\rightarrow X\). In this case, since
in \(\widetilde{\mathcal {A}}\), we see that a “complementary” summand of
in
has been added. This discrepancy has been noticed previously; see e.g. Henrard–van Roosmalen [18, Prop. A.11].
, given by
on objects, which is 2-universal among additive functors from \(\mathcal {A}\) to weakly idempotent complete categories; see e.g. [28, Rem. 1.12] or [11, Rem. 7.8].
; and, in addition,
, there exists a unique natural transformation
with
.
denote the inclusion functor of the subcategory \(\widehat{\mathcal {A}}\) into \(\widetilde{\mathcal {A}}\). The functor
factors through
as
. An additive functor \(\mathscr {F}:\widehat{\mathcal {A}} \rightarrow \mathcal {B}\) to a weakly idempotent complete category \(\mathcal {B}\) is determined up to unique natural isomorphism by its behaviour on the image
of \(\mathcal {A}\) in \(\widehat{\mathcal {A}}\); similarly, a natural transformation
of additive functors \(\widehat{\mathcal {A}}\rightarrow \mathcal {B}\) is also completely determined by its action on
; see [11, Rems. 6.7, 6.9].
, one must choose a kernel and an image of the idempotent \(\mathscr {F}(e)\) for each idempotent e in \(\mathcal {A}\).
is a biadditive functor.
the identity element of the abelian group \(\mathbb {E}(C,A)\). Suppose \(\delta \in \mathbb {E}(C,A)\) and that \(a:A\rightarrow B\) and \(d:D\rightarrow C\) are morphisms in \(\mathcal {C}\). We put
and
. Since \(\mathbb {E}\) is a bifunctor, we have that
.
.
be a product and
be a coproduct in \(\mathcal {C}\), and let \({\delta \in \mathbb {E}(C,A)}\) and \(\rho \in \mathbb {E}(D,B)\) be \(\mathbb {E}\)-extensions. The direct sum of \(\delta \) and \(\rho \) is the unique \(\mathbb {E}\)-extension \(\delta \oplus \rho \in \mathbb {E}(C\oplus D,A\oplus B)\) such that the following equations hold.
From the Yoneda Lemma, each \(\mathbb {E}\)-extension \(\delta \in \mathbb {E}(C,A)\) induces two natural transformations. The first is
given by
for all objects \(B\in \mathcal {C}\) and all morphisms \(a:A\rightarrow B\). The second is
and defined by
for all objects \(D\in \mathcal {C}\) and all morphisms \(d:D\rightarrow C\).
. If
, we depict
as 
be complexes, and suppose that
and
are \(\mathbb {E}\)-extensions. 
is known as an \(\mathbb {E}\)-attached complex if
and
. An \(\mathbb {E}\)-attached complex
is called an n-exangle (for \((\mathcal {C},\mathbb {E})\)) if, further, the sequences 
of \(\mathbb {E}\)-attached complexes is given by a morphism
such that
. Such an
is called a morphism of n-exangles if
and
are both n-exangles.
and
is the pair
.
of
in the following way. An object
is an object of
that satisfies
and
. For
, a morphism
is a morphism
with
and
. Note that this implies
is not necessarily a full subcategory of
, nor necessarily additive.
be complexes. Two morphisms in
are said to be homotopic if they are homotopic in the standard sense viewed as morphisms in
. This induces an equivalence relation \(\sim \) on
. We define
as the category with the same objects as
and with
.
is called a homotopy equivalence if its image in the category
is an isomorphism. In this case,
and
are said to be homotopy equivalent. The isomorphism class of
in
(equivalently, its homotopy class in
) is denoted
. Since the (usual) homotopy class of
in \(\textsf {\textrm{Ch}}({\mathcal {C}})\) may differ from its homotopy class in
, we reserve the notation
specifically for its isomorphism class in
.
the object in
given by
for \(j = i, i+1\) and
for \(0 \leqslant j \leqslant i-1\) and \(i+2 \leqslant j \leqslant n+1\), as well as
.
in
. The correspondence \(\mathfrak {s}\) is called an exact realisation of \(\mathbb {E}\) if it satisfies the following conditions. In case \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\) and
the following terminology is used. The morphism
is said to be an \(\mathfrak {s}\)-inflation and the morphism
an \(\mathfrak {s}\)-deflation. The pair
is known as an \(\mathfrak {s}\)-distinguished n-exangle.
and
, there exists
such that \(f_{0}=a\) and
. In this setting, we say that
realises \(\delta \) and
is a lift of (a, c).
, then
is an n-exangle.
and
.
. We will often use the diagram 
is an \(\mathfrak {s}\)-distinguished n-exangle. If we also have that
and
is a morphism of n-exangles, then we call
a morphism of \(\mathfrak {s}\)-distinguished n-exangles and we depict this by the following commutative diagram. 
is a morphism in
, such that
for some
. The mapping cone
of
is the complex 
,
, and
for \(i\in \{1,\ldots ,n-1\}\).
and an exact realisation \(\mathfrak {s}\) of \(\mathbb {E}\), such that the following conditions are met.
and
, then there exists a morphism
lifting
, such that
. In this case, the morphism
is called a good lift of
.
and obvious restrictions thereof. These are all defined in the usual way. However, by abuse of notation, we simply write \(\mathscr {F}\) for each of these.
. The pair \((\mathscr {F},\Gamma ) :(\mathcal {C},\mathbb {E},\mathfrak {s}) \rightarrow (\mathcal {C}',\mathbb {E}',\mathfrak {s}')\) is called an n-exangulated functor if, for all \(A,C\in \mathcal {C}\) and each \(\delta \in \mathbb {E}(A,C)\), we have that
whenever
.
This is an n-exangulated functor \((\mathcal {C},\mathbb {E},\mathfrak {s})\rightarrow (\mathcal {C}'',\mathbb {E}'',\mathfrak {s}'')\); see [9, Lem. 3.19(ii)].
and
are \(\mathfrak {s}\)-distinguished. 
is a morphism of \(\mathbb {E}\)-attached complexes, then the induced morphism
is a morphism of \(\mathbb {E}'\)-attached complexes.
as \(\mathbb {E}'\)-attached complexes.
since \(\Gamma \) is natural and
is an \(\mathbb {E}\)-attached complex. Similar computations show that both
and
are \(\mathbb {E}'\)-attached complexes. As
is a morphism
of complexes, it suffices to prove
This follows immediately from
and the naturality of \(\Gamma \).
of functors is said to be n-exangulated if, for all \(A,C\in \mathcal {C}\) and each \(\delta \in \mathbb {E}(C,A)\), we have
We denote this by
. In addition, if
has an n-exangulated inverse, then it is called an n-exangulated natural isomorphism. It is straightforward to check that
has an n-exangulated inverse if and only if
is an isomorphism for each \(X\in \mathcal {C}\).
the inclusion of the category \(\mathcal {C}\) into its idempotent completion \(\widetilde{\mathcal {C}}\); see Sect. 2.
(see Sect. 4.1) and an exact realisation \(\mathfrak {t}\) of \(\mathbb {F}\) (see Sect. 4.2), and then show that \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) is an n-exangulated category (see Sects. 4.3–4.5). For \(n=1\), we recover the main results of [27]. First, we establish some notation to help our exposition.
. Recall from Definition 2.3 that the identity of an object \((X,e)\in \widetilde{\mathcal {A}}\) is
.
as follows. For objects
and
in \(\widetilde{\mathcal {C}}\), we put
For morphisms
in \(\widetilde{\mathcal {C}}\), we define
and
, and an \(\mathbb {F}\)-extension
, we have Therefore,
lies in
. It is then straightforward to verify that \(\mathbb {F}\) is indeed a functor.
is an abelian group by defining
for
. The additive identity element of
is
. The inverse of
is
. Notice that we get an abelian group monomorphism:
This homomorphism plays a role later in the proof of Theorem 4.39.
, the pair
is a morphism of \(\mathbb {E}\)-extensions \(\delta \rightarrow \delta \). Indeed, we have that
from Definition 4.4.
by \({\tilde{\delta }}\). We call
the underlying \(\mathbb {E}\)-extension of \({\tilde{\delta }}\).
and
with
, there is a unique \(\mathbb {F}\)-extension
with underlying \(\mathbb {E}\)-extension \(\delta \). This \(\mathbb {F}\)-extension is
. Again, we use this instance of the tilde notation for this correspondence: we write
if and only if the underlying \(\mathbb {E}\)-extension of \({\tilde{\rho }}\) is \(\rho \), i.e.
.
are equal if and only if their underlying \(\mathbb {E}\)-extensions are equal. Hence, removing the tilde from \(\mathbb {F}\)-extensions defines an injective abelian group homomorphism \(\mathbb {F}((Y,e'), (X,e)) \rightarrow \mathbb {E}(Y,X)\) for \((X,e), (Y,e') \in \widetilde{\mathcal {C}}\).
be a complex in an additive category \(\mathcal {A}\) and suppose
is a weak cokernel of
. Suppose
is a morphism of complexes with
and
both idempotent. Then there exists a morphism
, such that the following hold. 
is a morphism of complexes.
is idempotent and satisfies
.
is an idempotent morphism of complexes.
is a homotopy of morphisms
, then the pair
yields a homotopy
.
as obtained in Lemma 4.9. Define \(q :=xp'_{2}\) and set
. We show this morphism satisfies the claims in the statement. For this, we will make use of the following. Let \(p=p(x)\in \mathbb {Z}[x]\) be any polynomial. Since
is a morphism of complexes, we have that
is also a morphism of complexes, i.e. the diagram 
, where the last equality follows from Lemma 4.9(i). Similarly,
. Thus, using \(p=q\) in the commutative diagram (4.1) shows that
is a morphism of complexes.
. Furthermore, we see that
. Thus, to show that
is idempotent, it is enough to show that \((f_1^2 - f_1)^2 = 0\) by Lemma 4.9(ii). Let \(r(x) = x^2 -x\). We see that
and
vanish as
and
are idempotents. Therefore, by choosing \(p = r\) in (4.1) we have
and so there is
with
, because
is a weak cokernel of
. This implies
as
, and hence
is idempotent.
is a homotopy. Then we see that
Hence,
is a null homotopy as desired. \(\square \)
is a possible choice for \(m=2\) in Lemma 4.9. Letting \(h = x^2 - x\) and \(i = x\), we see that
. Then the idempotent
obtained in Lemma 4.10 is the idempotent obtained through the idempotent lifting trick in [27, Lem. 3.5].
is an \(\mathfrak {s}\)-distinguished n-exangle and
is an idempotent with
. Then
can be extended to a null homotopic, idempotent morphism
with
for \(2 \leqslant i \leqslant n+1\). Further, the null homotopy of \(e_{\bullet }\) can be chosen to be of the shape
.
so
is a morphism of \(\mathbb {E}\)-extensions. The solid morphisms of the diagram 
making the two leftmost squares commute. Since
is an \(\mathfrak {s}\)-distinguished n-exangle, there is an exact sequence 
is in the kernel of
as
. Therefore, there exists
with
. If we put
, then
is morphism of \(\mathfrak {s}\)-distinguished n-exangles and
is a homotopy. By Lemma 4.10, using that
and 0 are idempotents, there is an idempotent
, such that
is an idempotent morphism of complexes and that
is a homotopy. Finally,
is a morphism of \(\mathfrak {s}\)-distinguished n-exangles since
. \(\square \)
and that
is an \(\mathfrak {s}\)-distinguished n-exangle. The morphism
of \(\mathbb {E}\)-extensions has a lift
that is idempotent and satisfies
for all \(2 \leqslant i \leqslant n -1\), such that there is a homotopy
.
and
, we have
and so
. Therefore, by Lemma 4.12 we can extend
of \(\mathfrak {s}\)-distinguished n-exangles with \(e'_{i} = 0\) for \(i\in \{ 2,\ldots ,n+1 \}\), having a homotopy
. Similarly, by the dual of Lemma 4.12, we can extend \(e''_{n+1}\) to an idempotent morphism
with \(e''_{i} = 0\) for \(i\in \{ 0,\ldots ,n-1 \}\), such that there is a homotopy
. Consider the morphism
and hence
is a homotopy.
and
is a homotopy. Lemma 4.10 yields an idempotent morphism
and a homotopy
. Then
and
are the desired idempotent morphism and homotopy, respectively.
is idempotent. Hence,
and
are the desired idempotent morphism and homotopy, respectively. \(\square \)
is a complex in \(\mathcal {C}\) and
is an idempotent morphism of complexes. We denote by
the complex in \(\widetilde{\mathcal {C}}\) with object
in degree i and differential
.
satisfies
since
is a morphism of complexes and consists of idempotents. Furthermore, whenever we write
to denote a complex in \(\widetilde{\mathcal {C}}\), we always mean that
is an idempotent morphism in \(\textsf {\textrm{Ch}}({\mathcal {C}})\) and that
is the induced object in \(\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})\) as described in Definition 4.15.
. Because of the need to tweak the differentials in
according to (4.2), one cannot recover the original complex
with differentials
from the pair
defined in Definition 4.15. This is in contrast to the description of an object in \(\widetilde{\mathcal {C}}\) as a pair (X, e) where one can recover \(X\in \mathcal {C}\) uniquely. Thus,
is an abuse of notation but should hopefully cause no confusion.
are objects in \(\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})\) and that
is a morphism in
. Then defining
for each \(i\in \mathbb {Z}\) gives rise to a morphism
in \(\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})\) with
.
is a morphism of complexes
. In this case, we call
the underlying morphism of
.
. Further, suppose that
is an \(\mathfrak {s}\)-distinguished n-exangle and
is an idempotent lift of
. Then
is an n-exangle for \((\widetilde{\mathcal {C}}, \mathbb {F})\).
and that
.
for some \(1\leqslant i \leqslant n\), suppose we have a morphism
with \(\tilde{d}_i^{(X,e)} \tilde{r} = 0\), that is,
. As
, we see that \(d_i^Xr=0\), whence there exists
such that \(d_{i-1}^Xs = r\) because
is an \(\mathfrak {s}\)-distinguished n-exangle. By Lemma 4.14, there is a morphism
with \(t = e_{i-1} s e'\). Then we observe that
, whence \(\tilde{d}_{i-1}^{(X,e)} \tilde{t} = \tilde{r}\).
is a morphism with
. Then we have
. Hence, there is a morphism
such that
as
is an \(\mathfrak {s}\)-distinguished n-exangle. Then the morphism
with
satisfies \(\tilde{d}^{(X,e)}_n \tilde{w} = \tilde{u}\), as required. \(\square \)
and that
in \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\). If
and
are idempotent lifts of
, then
are isomorphic in
, i.e.
.
and
are both n-exangles in \((\widetilde{\mathcal {C}}, \mathbb {F})\) by Lemma 4.18. Hence, we only have to show that
are both non-empty. Since we have
, there are morphisms
and
in
(with
and
). We then obtain morphisms
and
in
with
and
by Lemma 4.16. Note that
and
. So, since
, we have that
and \(\tilde{k}_{\bullet }\) are morphisms in
and we are done. \(\square \)
and
, and let \((\tilde{a},\tilde{c}):{\tilde{\delta }} \rightarrow {\tilde{\rho }}\) be a morphism of \(\mathbb {F}\)-extensions. Suppose
and
. Since (a, c) is a morphism of \(\mathbb {E}\)-extensions, there is a lift
of it using that \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\). As
and
are morphisms in \(\widetilde{\mathcal {C}}\), we have that
and
. Hence, by Lemma 4.16, it follows that
with
lifts \((\tilde{a},\tilde{c})\).
is an idempotent morphism lifting
. Thus, by Definition 4.20 and using
, we see that 
. \(\square \)
and an exact realisation \(\mathfrak {t}\) of \(\mathbb {F}\), we can begin to verify axioms (EA1), (EA2) and (EA2)
. In this subsection, we will check that the collection of \(\mathfrak {t}\)-inflations is closed under composition. One can dualise the results here to see that \(\mathfrak {t}\)-deflations compose to \(\mathfrak {t}\)-deflations.
. It is an analogue of [22, Lem. 2.1] for n-exangulated categories, allowing us to complete a “partial” lift of a morphism of extensions.
and
be \(\mathfrak {s}\)-distinguished n-exangles. Let l, r be integers with \(0 \leqslant l \leqslant r-2 \leqslant n-1\). Suppose there are morphisms
and
, where
, such that
is a morphism of \(\mathbb {E}\)-extensions and the solid part of the diagram 
such that (4.3) commutes.
is a morphism of \(\mathbb {E}\)-extensions. Now assume that the result holds for \(l=0\) and some \(3 \leqslant r\leqslant n+1\). Suppose we are given morphisms
and
such that
for \(i\in \{ r-1,\ldots , n \}\). By the induction hypothesis, we obtain a morphism 
. Next, note that we have
. Since
is a weak kernel of
, there exists
so that
. Set
for
. Notice that we have
for \(i \notin \{r-1, r-2\}\). We claim that (4.3) commutes. By construction, we only need to check commutativity of the two squares involving
. These indeed commute since
and
using the commutativity of (4.4). This concludes the base case \(l=0\).
is a weak cokernel of
. \(\square \)
is an \(\mathfrak {s}\)-distinguished n-exangle. Assume
and
are idempotents, such that
and 
and
can be extended to an idempotent morphism
with
for
.
such that
is a morphism of \(\mathfrak {s}\)-distinguished 1-exangles. Recall the polynomial
from Lemma 4.9. We will show that
, where
, is the desired idempotent morphism of \(\mathfrak {s}\)-distinguished n-exangles.
is an \(\mathfrak {s}\)-distinguished 1-exangle, there is an exact sequence 
, there exists a morphism
with
. This shows that
because
is idempotent. Hence,
is an idempotent morphism of complexes by Lemma 4.9(ii). Furthermore,
using Lemma 4.9(i), so that
is a morphism of \(\mathbb {E}\)-extensions. This computation also shows the existence of
with underlying \(\mathbb {E}\)-extension \(\delta \).
. Therefore,
is an element of
with underlying \(\mathbb {E}\)-extension \(\delta \). The solid morphisms of the diagram 
is a morphism of \(\mathbb {E}\)-extensions as
. Since the rows are the \(\mathfrak {s}\)-distinguished n-exangle
, by Lemma 4.23 we can find a morphism
, so that the diagram above is a morphism
. Furthermore, as
and
are idempotent, we may assume that
is an idempotent by Lemma 4.10. \(\square \)
, we cannot a priori say too much about how
might look. This is one of the main issues in trying to prove (EA1) for \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\). The next lemma gives us a way to deal with this and is the last preparatory result we need before the main result of this subsection.
be a \(\mathfrak {t}\)-inflation. Then there is an \(\mathfrak {s}\)-distinguished n-exangle \(\langle X'_{\bullet }, \delta \rangle \) with
and
for some \(C \in \mathcal {C}\), such that
and
for some
.
is an \(\mathfrak {t}\)-inflation, there is a \(\mathfrak {t}\)-distinguished n-exangle
with
,
and \(\tilde{d}_0^{\widetilde{Y}} = \tilde{f}\). By definition of \(\mathfrak {t}\), this means there is an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y'_{\bullet }, \delta ' \rangle \) with an idempotent morphism \(e'_{\bullet } :\langle Y'_{\bullet }, \delta ' \rangle \rightarrow \langle Y'_{\bullet }, \delta ' \rangle \), such that
and
, and there are mutually inverse homotopy equivalences
which satisfy
and
. Note that we, thus, have
and
. In particular, we have a commutative diagram 
.
and the \(\mathbb {E}\)-extension
. Note that if \(n=1\), then
; otherwise we have \({{Y''_{n+1}}} =0\). In either case, we have an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y''_{\bullet }, \delta '' \rangle \) using the axiom (R2) for \(\mathfrak {s}\), and hence also an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y''_{\bullet } \oplus Y'_{\bullet }, \delta '' \oplus \delta ' \rangle \) by [16, Prop. 3.3]. Using the canonical isomorphism
we see that the complex 
in \((\mathcal {C},\mathbb {E},\mathfrak {s})\) by [16, Cor. 2.26(2)]. Consider the diagram 
and
. This diagram commutes since
and
Notice that the composition ba is an automorphism of
, and so the complex 
as
,
and
. Setting
and \(C :=Y'_{1}\) finishes the proof. \(\square \)
and
are \(\mathfrak {t}\)-inflations with
. Then
is a \(\mathfrak {t}\)-inflation.
and
for some \(C \in \mathcal {C}\), so that
for some
and
. Similarly, there is also an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Y'_{\bullet }, \delta ' \rangle \) with
and
for some \(C' \in \mathcal {C}\), so that
for some
and
. Setting
, we also have the n-exangle
by axiom (R2) for \(\mathfrak {s}\). Then
is \(\mathfrak {s}\)-distinguished by [16, Prop. 3.3]. We have
and
, and
is the \(\mathfrak {s}\)-inflation of
with respect to the given decompositions. Since
and \(d^{Y'\oplus Y''}_0\) are \(\mathfrak {s}\)-inflations, by (EA1) for \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\), we have that the morphism
is an \(\mathfrak {s}\)-inflation, where we used that
. Therefore, there is an \(\mathfrak {s}\)-distinguished n-exangle \(\langle Z''_{\bullet }, \delta '' \rangle \) with
,
and
.
. Since
, we can apply [16, Prop. 3.6(1)] to obtain a morphism 
As
and
is idempotent, we see that
This implies that
Since \(\langle X'_{\bullet }, \delta \rangle \) is an \(\mathfrak {s}\)-distinguished n-exangle, by [16, Lem. 3.5] there is an exact sequence 
vanishes under
, so there is a morphism
with
. Since
, this implies
showing that
.
. A quick computation yields the equality
. Therefore, by Lemma 4.24, there is an idempotent morphism \(e''_{\bullet } :\langle Z''_{\bullet }, \delta '' \rangle \rightarrow \langle Z''_{\bullet }, \delta '' \rangle \) with
, \(e''_{1} = e'_{1} \oplus 0 \oplus 0\) as well as an \(\mathbb {F}\)-extension
with underlying \(\mathbb {E}\)-extension \(\rho = \delta ''\). We obtain a \(\mathfrak {t}\)-distinguished n-exangle \(\langle (Z''_{\bullet }, e''_{\bullet }), {\tilde{\rho }} \rangle \). Then the \(\mathfrak {t}\)-inflation of this n-exangle is given by the morphism
satisfying
As
is an isomorphism in \(\widetilde{\mathcal {C}}\), the complex 
and
forms part of the \(\mathfrak {t}\)-distinguished n-exangle \(\langle \widetilde{X}''_{\bullet }, {\tilde{\rho }} \rangle \) by [16, Cor. 2.26(2)]. \(\square \)
also holds. We need two key technical lemmas first.
of the morphism
of \(\mathbb {E}\)-extensions exists, so that 
as morphisms
.
is an \(\mathbb {F}\)-extension;
is a morphism in \(\widetilde{\mathcal {C}}\) for some
;
and
are \(\mathfrak {s}\)-distinguished n-exangles with
;
is an idempotent morphism lifting
, such that
is null homotopic; and
is an idempotent morphism lifting
, such that
is null homotopic.
and \( c e'_{n+1}= c\) imply
Therefore,
is indeed a morphism of \(\mathbb {E}\)-extensions and condition (v) makes sense. Condition (iv) makes sense due to Remark 4.5(iv).
is a morphism of \(\mathbb {E}\)-extensions, it admits a good lift
using axiom (EA2) for the n-exangulated category \((\mathcal {C},\mathbb {E},\mathfrak {s})\). Define
for \(0\leqslant i \leqslant n+1\). Note that
and
by assumption. For \(i=0\), we have
. On the other hand, for \(i=n+1\) we have
. Therefore, the morphism
is of the form
.
and
, respectively, are morphisms of complexes. The squares on the front and back faces in (4.7) commute because
is the sum of morphisms of complexes from
to
. This also implies
is a lift of
. Of the remaining squares, the leftmost clearly commutes and the rightmost commutes as
is a morphism in \(\widetilde{\mathcal {C}}\). For \(1\leqslant i \leqslant n\), we have
Therefore, diagram (4.7) commutes and, further, the last assertion follows.
is a good lift of
. Recall that
and
are both null homotopic by assumption, and so
is also null homotopic. Then it follows from [16, Rem. 2.33(1)] that
is a good lift of
since \(g'_{\bullet }\) is. \(\square \)
with under-lying morphism
is a good lift of the morphism
of \(\mathbb {F}\)-extensions.
is indeed an \(\mathbb {F}\)-extension and
a morphism of \(\mathbb {F}\)-extensions. Using
and
, as well as the commutativity of (4.7), we see that
is a morphism
of \(\mathfrak {t}\)-distinguished n-exangles, lifting
.
denotes the mapping cone of
in \(\mathcal {C}\), and that
is \(\mathfrak {s}\)-distinguished as
is a good lift of
. Using the commutativity of (4.7), that
and
are morphisms of complexes, and that
, one can verify that the diagram 
This calculation also shows that
. Thus, by definition of \(\mathfrak {t}\), we have that \(\mathfrak {t}({\tilde{\rho }}) = [ ({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet }) ]\), i.e. \(\langle ({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet }), {\tilde{\rho }} \rangle \) is \(\mathfrak {t}\)-distinguished.
in
, so \(\langle {M^{\widetilde{\mathcal {C}}}_{\tilde{h}}}_{\bullet }, {\tilde{\rho }} \rangle \) is \(\mathfrak {t}\)-distinguished. Lastly, we note that
because
Hence,
is a \(\mathfrak {t}\)-distinguished n-exangle. \(\square \)
can be shown dually.
-extension and suppose
is a morphism in \(\widetilde{\mathcal {C}}\). Suppose
are \(\mathfrak {t}\)-distinguished n-exangles. Then
has a good lift
.
is
. By definition of \(\mathfrak {t}\) and Remark 4.21, there are \(\mathfrak {s}\)-distinguished n-exangles \(\langle X'_{\bullet }, \delta \rangle \) and
and idempotent morphisms
and
, such that
and \(\mathfrak {t}(\tilde{c}^\mathbb {F}{\tilde{\delta }}) = [(Y'_{\bullet }, e'_{\bullet })]\), and so that
and
are null homotopic in
. We note that since
and
, we have that
and, in particular, that
. Moreover, it follows that all the hypotheses of Lemma 4.27 are satisfied.
is semi-perfect since
is fully faithful (see Proposition 2.5). By Remark 2.6, we have that
. In particular, we see that
is an idempotent subring of the semi-perfect ring
. Hence, by Anderson–Fuller [1, Cor. 27.7], we have that the endomorphism ring of each object in \(\widetilde{\mathcal {C}}\) is semi-perfect. \(\square \)
be a \(\mathfrak {t}\)-inflation and suppose there is a morphism
in \(\widetilde{\mathcal {C}}\) with \(\tilde{f} \tilde{g} = \widetilde{0}\). By Lemma 4.25, there is an \(\mathfrak {s}\)-inflation
, which is monic as \((\mathcal {C}, \mathbb {E}, \mathfrak {s})\) is n-exact. We have \(fg = 0\) as \(\tilde{f} \tilde{g} = \widetilde{0}\), and we also have
because the underlying morphism g of \(\tilde{g}\) satisfies
. Thus, we see that
and this implies \(g = 0\) as
is monic. Hence, \(\tilde{g} = 0\) and we are done. \(\square \)
. We will show that
forms part of an n-exangulated functor
, and that this is 2-universal in an appropriate sense. The next lemma is straightforward to check.
for
, defines a natural isomorphism
.
is an n-exangulated functor from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\).
is an \(\mathfrak {s}\)-distinguished n-exangle, then
is \(\mathfrak {t}\)-distinguished, where
. We have the idempotent morphism
, so from Definition 4.20 we see that
. \(\square \)
by Remark 2.6. By
and
, we denote the canonical inclusion and projection morphisms, respectively.
, the \(\mathbb {E}'\)-attached complexes and morphisms between them were defined in Definition 3.1, and together they form an additive category.
be an \(\mathbb {F}\)-extension. Suppose
and
is an idempotent morphism. With
, we have that
as \(\mathfrak {t}\)-distinguished n-exangles.
. First, note that
and
are \(\mathbb {F}\)-attached complexes, and
is a \(\mathfrak {t}\)-distinguished n-exangle since \((\mathscr {F},\Gamma )\) is an n-exangulated functor.
and
of complexes induced by
, as well as the corresponding ones \(\tilde{i}_{e'_{\bullet }}\) and \(\tilde{p}_{e'_{\bullet }}\) for \(e'_{\bullet }\). We claim that there is a biproduct diagram 
is a morphism of \(\mathbb {F}\)-attached complexes, we just need to check that
. By Remark 4.7, it is enough to see that
holds, and this is indeed true because
is a morphism of \(\mathbb {F}\)-attached complexes. To see that \(\tilde{i}_{e'_{\bullet }}\) and \(\tilde{p}_{e'_{\bullet }}\) are morphisms of \(\mathbb {F}\)-attached complexes, one uses that
. Furthermore, we have the identities
,
and
, so (4.11) is a biproduct diagram in the additive category of \(\mathbb {F}\)-attached complexes. Therefore, we have that (4.10) is an isomorphism as \(\mathbb {F}\)-attached complexes.
is 2-universal amongst n-exangulated functors from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to idempotent complete n-exangulated categories.
.
, there is a unique n-exangulated natural transformation
with
.
. It remains to show that \(\mathscr {E}\) forms part of an n-exangulated functor \((\mathscr {E},\Psi )\) and that
is n-exangulated.
, we set
For
and
in \(\widetilde{\mathcal {C}}\), we define an abelian group homomorphism
and put
For morphisms
and
in \(\mathcal {C}\), we have
as
is natural. For morphisms
, we have
using how \(\mathbb {F}\) is defined on morphisms (see Definition 4.4). We claim that the family of abelian group homomorphisms
for
defines a natural transformation \(\Psi :\mathbb {F}(-,-) \Rightarrow \mathbb {E}'(\mathscr {E}-, \mathscr {E}-)\). To this end, fix objects
and
, and morphisms
and
in \(\widetilde{\mathcal {C}}\). First, note that we have
and, similarly,
Therefore, we see that
Next, we must show that \((\mathscr {E},\Psi )\) sends \(\mathfrak {t}\)-distinguished n-exangles to \(\mathfrak {s}'\)-distinguished n-exangles. Thus, let
and
, and suppose
. We need that
, which will follow from seeing that
is a direct summand of an \(\mathfrak {s}'\)-distinguished n-exangle.
in
with
and an idempotent morphism
lifting
, such that
. Note for later that we thus have
, and hence
. Let
. Since \((\mathscr {F}, \Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) is an n-exangulated functor, the n-exangle
is \(\mathfrak {s}'\)-distinguished. As we have an isomorphism of complexes
, the \(\mathbb {E}'\)-attached complex
is \(\mathfrak {s}'\)-distinguished by [16, Cor. 2.26(2)]. Since is just the identity homomorphism, a quick computation yields
In particular, this implies that
is \(\mathfrak {s}'\)-distinguished.
as \(\mathbb {F}\)-attached complexes by Lemma 4.38, where
. We see that
is a direct summand of the \(\mathfrak {s}'\)-distinguished n-exangle
by Proposition 3.7(ii). Hence,
by [16, Prop. 3.3], and so \((\mathscr {E},\Psi )\) is an n-exangulated functor.
with
, so it remains to show that
induces an n-exangulated natural transformation \((\mathscr {E}, \Psi ) \Rightarrow (\mathscr {G}, \Theta )\). For this, let
and
be arbitrary. Note that we have
Hence, we obtain
and the proof is complete. \(\square \)
. Recall that in this case \(\mathbb {E}(Z,X) = \mathcal {C}(Z,\Sigma X)\) for \(X,Z\in \mathcal {C}\). Using [25, Thm. 3.1], one obtains an \((n+2)\)-angulated category
, where \(\widetilde{\Sigma }\) is induced by \(\Sigma \). From this \((n+2)\)-angulated category, just like above, we obtain an induced n-exangulated category \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\). Notice that \(\mathbb {F}'(-,-) = \widetilde{\mathcal {C}}(-,\widetilde{\Sigma }-)\). Comparing \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\) to the n-exangulated category \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) found from Theorem 4.32, again we see that \(\mathbb {F}\) and \(\mathbb {F}'\) differ by the labelling convention we chose in Definition 4.4. By [16, Prop. 4.8] we have that \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) induces an \((n+2)\)-angulated category
, and therefore the n-exangulated inclusion functor
is, moreover, \((n+2)\)-angulated. It follows from [25, Thm. 3.1(2)] that
and
must be equal, and hence \((\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})\) and \((\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')\) are n-exangulated isomorphic.
are straightforward to check directly. It is then clear that \(\mathcal {A}\) inherits an n-exangulated structure from \(\mathcal {C}\).
of \(\mathcal {C}\) into \(\widetilde{\mathcal {C}}\) is part of an n-exangulated functor
, which satisfies the 2-universal property from Theorem 4.39. In this section, we turn our attention to the weak idempotent completion \(\widehat{\mathcal {C}}\) of \(\mathcal {C}\) and we show that it forms part of a triplet \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\) that is n-extension-closed (see Definition 5.2) in \((\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})\). It will then follow that \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\) is itself n-exangulated, and, moreover, there is an analogue of Theorem 4.39 for \((\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})\); see Theorem 5.5.
are objects,
is an \(\mathbb {F}\)-extension and
. Then there is a \(\mathfrak {t}\)-distinguished n-exangle
with
and an \(\mathfrak {s}\)-distinguished n-exangle
, such that
as \(\mathfrak {t}\)-distinguished n-exangles.
with
for \(2 \leqslant i \leqslant n-1\), as well as a homotopy
, where
. Notice
for \(i = 0, n+1\) by assumption. Furthermore,
for
. By Lemma 4.38 we have
-distinguished n-exangles. We will show that there is an isomorphism
in
for some
, as well as an isomorphism
in
for some object
.
is split by assumption, so by Lemma 2.4 there are objects \(Y'_i \in \mathcal {C}\) and isomorphisms
. For \(2\leqslant i \leqslant n-1\), we see that
, so by Lemma 2.4 again we have isomorphisms
, but now where \( Y'_{i} = 0 \in \mathcal {C}\). Since
by assumption and because
for \(2 \leqslant i \leqslant n-1\), we put
and
for \(i \in \{ 0,n+1\} \cup \{2,\ldots , n-1\}\). It remains to find appropriate isomorphisms
and \(\tilde{s}'_i\) for \(i=1,n\).
with underlying morphism
and another
with underlying morphism
by Lemma 4.14. Since
is a homotopy, we see that
. This implies
and so
. Similarly, we also have
. The complex
with object
in degree \(0\leqslant i \leqslant n+1\) and differential
in degree \(0\leqslant i\leqslant n\) is isomorphic to
via
. Furthermore, as
and
are identity morphisms, we have that
is \(\mathfrak {t}\)-distinguished by [16, Cor. 2.26(2)]. The complex \(\widetilde{Y}'_{\bullet }\) with object
in degree \(0\leqslant i \leqslant n+1\) and differential
in degree \(0\leqslant i \leqslant n\) is isomorphic to
via \(\tilde{s}'_{\bullet }\). Moreover, this induces an isomorphism
of \(\mathbb {F}\)-attached complexes, and hence of \(\mathfrak {t}\)-distinguished n-exangles. It is clear that
by the construction of \(\widetilde{Y}'_\bullet \). Lastly,
is \(\mathfrak {s}\)-distinguished using [16, Prop. 3.3] and that \(\mathfrak {s}\) is an exact realisation of \(\mathbb {E}\). \(\square \)
is a section in the complex
, and hence this complex is a split short exact sequence by [16, Claim 2.15]. In particular, we have that
. So we put \(Y'_{1} :=Y'_{0}\oplus Y'_{2}\) and define
to be this composition of isomorphisms. As
, and
and
are isomorphic to objects in \(\widehat{\mathcal {C}}\), by Lemma 2.12 there is an isomorphism
for some
.
implies that the identities
and
hold. Therefore, we see that
which shows that
and
are mutually inverse isomorphisms. We now define \(Y'_{1} :=Y'_{0}\) and
. Because there are isomorphisms
, and
and
are isomorphic to objects in \(\widehat{\mathcal {C}}\), by Lemma 2.12 there is an isomorphism
for some
. In a similar way, one can show that
and
are mutually inverse isomorphisms. We set \(Y'_{\textit{n}} :=Y'_{n+1}\) and
. In addition, there is an isomorphism
for some
.
such that
for all \(1\leqslant i \leqslant n\) and
.
be the restriction of
.
, there is a \(\mathfrak {t}\)-distinguished n-exangle
with
by Proposition 5.1. We put
, the isomorphism class of
in
.
hold for the triplet \((\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})\). The difficult part is then to show that (EA1) is satisfied; this follows from Lemma 5.4 below. We note here, however, it has been shown in [23, Thm. A] that any n-extension-closed subcategory of an n-exangulated category that is also closed under isomorphisms inherits an n-exangulated structure in the expected way. Although the isomorphism-closure in [23] is assumed only for convenience, we highlight that the weak idempotent completion is not necessarily closed under isomorphisms in the idempotent completion using the constructions in Sect. 2. Indeed, one can show that \(\widehat{\mathcal {C}}\) is isomorphism-closed in \(\widetilde{\mathcal {C}}\) if and only if \(\mathcal {C}\) is already weakly idempotent complete.
be a \(\mathfrak {t}\)-inflation with
. Then there is a \(\mathfrak {t}\)-distinguished n-exangle
with
and
.
and an \(\mathfrak {s}\)-distinguished n-exangle
and
. The solid morphisms of the diagram 
,
and
for \(3 \leqslant i \leqslant n+1\), which makes the diagram above commute. Let
and
. Notice that the underlying \(\mathbb {E}\)-extension of \({\tilde{\rho }}'\) is \(\rho \). Set
. Then
as \(\mathfrak {t}\)-distinguished n-exangles by Lemma 4.38.
realises the trivial \(\mathbb {F}\)-extension
, we have that
is a section by [16, Claim 2.15]. If \(n=1\), then this is enough to see that (5.2) is split short exact. For \(n\geqslant 2\) we notice that there is an isomorphism
in \(\widetilde{\mathcal {C}}\), so
. Thus, since
is a weak kernel of
, we see that
factors through
. In particular, this implies
is a cokernel of
. Again, (5.2) is split short exact.
. We know that the objects
and
are isomorphic to objects in
by Lemma 2.4, as
and
are split idempotents. This implies that
is isomorphic to an object in \(\widehat{\mathcal {C}}\) by Lemma 2.12. Again Lemma 2.12 and the isomorphism
imply that there is an isomorphism
for some
.
with underlying morphism
is an isomorphism. Finally, put
for \(i = 0\) and \(3 \leqslant i \leqslant n+1\). Then the complex 
via
in
. With
, we see that
is \(\mathfrak {t}\)-distinguished by [16, Cor. 2.26(2)], as desired. \(\square \)
is an n-exangulated functor, such that the following 2-universal property is satisfied. Suppose \((\mathscr {F},\Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')\) is an n-exangulated functor to a weakly idempotent complete n-exangulated category \((\mathcal {C}', \mathbb {E}', \mathfrak {s}')\). Then the following statements hold. 
.
, there is a unique n-exangulated natural transformation
with
.
is an n-exangulated functor as in Proposition 4.36, by using the definition of \(\mathfrak {r}\) and noting that
lies in \(\widehat{\mathcal {C}}\) for all \(X\in \mathcal {C}\).
of \(\mathfrak {t}\)-distinguished n-exangles from the statement of Proposition 5.1 induces an isomorphism
of \(\mathfrak {r}\)-distinguished n-exangles.