So let
\({\mathsf {M}}\) be a symmetric monoidal category in which equalizers exist and are preserved by taking the monoidal product with any object. Let
\({\mathsf {C}}\) be the monoidal category of comonoids in
\({\mathsf {M}}\) and consider the monoidal admissible class
\({\mathcal {S}}\) of spans in
\({\mathsf {C}}\) from ([
3], Example 2.3). Since
\(\varepsilon \) is the counit, the diagram
commutes for any bimonoid morphism
g proving that
From Example 3.3 and Proposition 3.7 in [
3] we know that all
\({\mathcal {S}}\)-relative pullbacks exist in the category of bimonoids in
\({\mathsf {M}}\); hence any simplicial bimonoid in
\({\mathsf {M}}\)—that is, any functor from
\(\Delta ^{\mathsf {op}}\) to the category of monoids in
\({\mathsf {C}}\)—satisfies the successive assumptions of Assumption
2.1 for any positive integer. Still—say, for an easier comparison with [
8]—below we present a more explicit description of the objects
\(S_n^{(k)}\).
For any positive integer
n and any
\(0< k \le n\) the desired objects
\(S_n^{(k)}\) are constructed as the joint kernels of the morphisms
\(\{\partial _n,\partial _{n-1},\cdots , \partial _{n-k+1}\}\) in the category of bimonoids in
\({\mathsf {M}}\); that is, as the joint equalizers
in
\({\mathsf {M}}\) (where the “hat notation” of (
2) is used). By construction they are bimonoids. Using the universality of the equalizer (in
\({\mathsf {M}}\)) in the bottom rows, for
\(k>1\) we construct bimonoid morphisms in
(note the serial commutativity of the right-hand diagram thanks to the simplicial identities) and show that they give rise to the
\({\mathcal {S}}\)-relative pullback
From (
3) we infer
. The square of (
4) commutes since
\(j_{n-1}^{(k-1)}\) in the right vertical side of the commutative diagram
is a monomorphism. In order to check the universality of (
4), take a bimonoid morphism
such that the exterior of the left-hand diagram of
commutes (we know that
by (
3)). Then a filler
\(\widetilde{g}\) of the left-hand diagram of (
5) is constructed using the universality of the equalizer in the bottom row of the right-hand diagram of (
5). The occurring morphism
\(j_n^{(k-1)}\,\cdot \,g\) renders commutative the diagrams
for
\(n-k+1<i\le n\). Thus since it is a comonoid morphism, it equalizes the parallel morphisms of the right-hand diagram of (
5). The so constructed morphism
\(\widetilde{g}\) renders commutative the left-hand diagram of (
5) since the right column and the bottom row of the left-hand commutative diagram in
are equal monomorphisms. Finally
\(\tilde{g}\) is the unique filler of the left-hand diagram of (
5); as if also
h makes the left-hand diagram of (
5) commute then also the right-hand diagram of (
6) commutes. Since
\(j_n^{(k)}\) is a monomorphism, this proves
\(h=\widetilde{g}\). In order to see that the span (
4) satisfies the reflection property of ([
3], Definition 3.1) on the right, take bimonoid morphisms
such that
; equivalently, the large square on the left of the diagram
commutes. Since
is a monomorphism, this is equivalent to the commutativity of the exterior diagram. Since also
is a monomorphism, this is further equivalent to
\(hg\,\cdot \,c\,\cdot \,\delta =hg\,\cdot \,\delta \); that is, to
. Reflectivity on the left follows symmetrically.