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2015 | OriginalPaper | Chapter

8. Categories

Author : Jürgen Jost

Published in: Mathematical Concepts

Publisher: Springer International Publishing

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Abstract

This chapter will introduce the abstract and universal constructions of category theory.

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previous chapter Structures
next chapter Topoi
Footnotes
1
Some history about how those names evolved can be found in [82], and of course, this development possesses some internal consistency that may serve as a justification for the choice of terminology.
 
2
Denoting that object by 1 is not meant to carry any implications; in particular, this neither means a terminal object nor an identity element. It is simply an arbitrary label, like others used below, including \(0,2,3,\ldots \)
 
3
The name “exponential” seems to derive from the notation employed. It has little to do with the exponential function in analysis or the exponential map in Riemannian geometry and Lie groups. Perhaps a small justification is the following. When B and C are finite sets with n and m elements, resp., then the set \(C^B\) has \(m^n\) elements.
 
4
In order to facilitate the reading for those who are not accustomed to the abstract language employed in category theory, we try to spell out all the details. Therefore, more advanced readers can skip much of what follows below.
 
5
The scalar product \(\langle .,.\rangle :H\times H \rightarrow {\mathbb R}\) is symmetric, bilinear, and positive definite, that is,
$$\begin{aligned} \langle v,w\rangle = \langle w,v\rangle&\text { for all }v,w \in H\\ \langle \alpha v_1 +\beta v_2,w\rangle = \alpha \langle v_1,w\rangle + \beta \langle v_2,w\rangle&\text { for all }\alpha , \beta \in {\mathbb R}, v_1,v_2,w \in H\\ \langle v, v\rangle >0&\text { for all }v\ne 0 \in H.\\ \end{aligned}$$
.
 
Metadata
Title
Categories
Author
Jürgen Jost
Copyright Year
2015
Book
Mathematical Concepts

Print ISBN: 978-3-319-20435-2

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

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-20436-9_8

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