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2015 | OriginalPaper | Buchkapitel

2. Foundations

verfasst von : Jürgen Jost

Erschienen in: Mathematical Concepts

Verlag: Springer International Publishing

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Abstract

A set is a collection of distinct or distinguishable objects, its elements. But how can these elements be distinguished? Possibly, by certain specific intrinsic properties that they possess in contrast to others. Better, by specific relations that they have with other elements. In this chapter, we introduce the fundamental relations, operations and structures that will appear in this book.

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Vorheriges Kapitel Overview and Perspective

Nächstes Kapitel Relations

Please do not get confused by the different meanings of the symbols 0 and 1 here, as algebraic symbols on one hand and as members of a set on the other hand.

This is, of course, an example of a vector space, but that latter concept will only be introduced below. Therefore, we recall some details here.

Carefully distinguish the different meanings of the symbol 0 employed here!

In Definition 2.1.16, we have explained what a subgroup is, and you should then easily be able to define a subring, if you do not already know that concept.

Thus, we have equipped the group \({\mathbb Z}_2\) now with an additional operation, multiplication, but still denote it by the same symbol. We shall, in fact, often adopt the practice of not changing the name of an object when we introduce some additional structure or operation on it. That structure or operation will then henceforth be implicitly understood. This is a convenient, but somewhat sloppy practice. Probably, you will not need to worry about it, but as a mathematician, I should at least point this out.

Alternatively, as noted above, for an algebraic structure like a group, we could consider that structure as the single object, and its elements as the morphisms of that object.

Alternatively, we could also consider the elements of a group or monoid as the objects of the corresponding category. The morphisms would again be the multiplications by elements. Thus, the classes of objects and morphisms would coincide.

An obvious definition: A category \(\mathbf {D}\) is a subcategory of the category \(\mathbf {C}\) if every object D and every morphism \(D_1 \rightarrow D_2\) of \(\mathbf {D}\) is also an object or a morphism, resp., of \(\mathbf {C}\).

In fact, these form more naturally so-called bicategories. We suppress this technical point here, however. See for instance [80]. More importantly, one has to be careful to avoid paradoxes of self-reference. Therefore, one respects the axioms of set theory as listed in Sect. 2.2 and considers only sets from a universe U. A category will be called small if both its objects and its arrows constitute a set from U (see Definition 8.1.2). One then looks only at categories of small categories.

We refer to [44] for a conceptual analysis of Cuvier’s position in the history of biology.

see Sect. 3.4.1 for the definition of a cycle.

With certain exceptions that need not concern us here, all the cells of a given organism carry the same DNA sequence.

We assume here that there is a one-to-one correspondence between the loci of different members of the species. That is, we assume that the only differences between individuals are given by point mutations, but not by insertions or deletions of nucleotide strings in the genome.

Here, we can use the distance induced by the Fisher metric on the space of measures. We can also utilize the Kullback-Leibler divergence, which is not quite a distance, in fact, because it is not symmetric. For the definition and for a geometric view of these distances, see [3, 6].

Titel: Foundations
verfasst von: Jürgen Jost
Verlag: Springer International Publishing
Buch: Mathematical Concepts
Print ISBN: 978-3-319-20435-2

Electronic ISBN: 978-3-319-20436-9

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-20436-9_2

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