Functor category

In mathematical branch of category theory one is functor a category whose objects functors and their morphisms natural transformations are between these functors.

introduction

If and are two categories, then the natural transformations between functors behave like the morphisms of a category. Two natural transformations and between functors can be concatenated to form a natural transformation , so that the associative law applies to this concatenation , and for each functor there is an identical transformation which behaves like a neutral element in this concatenation . It therefore makes sense to consider the category of all functors with the natural transformations as morphisms. However, set- theoretical obstacles stand in the way of this, because a functor as a mapping between objects and morphisms of the source and target categories are generally not sets themselves, so they cannot be elements of a class . The same applies to the natural transformations between two functors and the “ power ” of the natural transformations between two functors is too great. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {D}}}$ ${\displaystyle \alpha :F\rightarrow G}$ ${\displaystyle \beta :G\rightarrow H}$ ${\displaystyle F,G,H:{\mathcal {C}}\rightarrow {\mathcal {D}}}$ ${\displaystyle \beta \circ \alpha :F\rightarrow H}$ ${\displaystyle F:{\mathcal {C}}\rightarrow {\mathcal {D}}}$ ${\displaystyle 1_{F}=(1_{F(C)})_{C\in {\mathcal {C}}}:F\rightarrow F}$ ${\displaystyle {\mathcal {D}}^{\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}\rightarrow {\mathcal {D}}}$ ${\displaystyle \alpha =(\alpha _{C}:F(C)\rightarrow G(C))_{C\in {\mathcal {C}}}}$ ${\displaystyle F}$ ${\displaystyle G}$

There are basically two ways out. You can avoid the set-theoretical problems with new terms, but then you have to be careful with formulations, or you limit yourself to small categories . ${\displaystyle {\mathcal {C}}}$

Quasi-categories

(The term quasi -category that follows is not used consistently in the literature; some authors also understand this term to include infinite categories that have nothing to do with the definition presented here.)

Families of classes are called conglomerates and it is said that a quasi-category consists of conglomerates and , whose "elements" are called objects or morphisms, and functions that assign each morphism its source or target area, as well as a mapping ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \mathrm {dom} ,\mathrm {cod} :{\mathcal {M}}\rightarrow {\mathcal {O}}}$

${\displaystyle \circ :D:=\{(f,g)\in {\mathcal {M}}\times {\mathcal {M}}\mid \mathrm {dom} (f)=\mathrm {cod} (g)\}\rightarrow {\mathcal {M}}}$, so that

(1): For applies and ${\displaystyle (f,g)\in D}$ ${\displaystyle \mathrm {dom} (f\circ g)=\mathrm {dom} (g)}$ ${\displaystyle \mathrm {cod} (f\circ g)=\mathrm {cod} (f)}$

(2): For applies ${\displaystyle (f,g),(g,h)\in D}$ ${\displaystyle f\circ (g\circ h)=(f\circ g)\circ h}$

(3): For each there exists such that and ${\displaystyle A\in {\mathcal {O}}}$ ${\displaystyle e_{A}\in {\mathcal {M}}}$ ${\displaystyle \mathrm {dom} (e_{A})=\mathrm {cod} (e_{A})=A}$

(a): for everyone with , ${\displaystyle f\circ e_{A}=f}$ ${\displaystyle f\in {\mathcal {M}}}$ ${\displaystyle \mathrm {dom} (f)=A}$

(b): for everyone with . ${\displaystyle e_{A}\circ f=f}$ ${\displaystyle f\in {\mathcal {M}}}$ ${\displaystyle \mathrm {cod} (f)=A}$

This allows the quasi-category of all functors to be formed with the natural transformations as morphisms; this includes the sub-quasi-category of all functors between two specified categories as above. Obviously, categories are also quasi-categories, so that this is a real generalization.

With the use of the name “conglomerate” the set-theoretical obstacles are of course not removed. Statements about quasi-categories must always be translated into "for all classes with a certain property ...".

Functor categories of small categories

If there is a small category in the introduction , the set theoretical problems do not exist and is a real category. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}^{\mathcal {C}}}$

A simple example is the category with two objects, such as and , and a single morphism different from identities . Then is nothing but the arrow category of . ${\displaystyle {\mathcal {C}}=2}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0\rightarrow 1}$ ${\displaystyle {\mathcal {D}}^{\mathcal {C}}={\mathcal {D}}^{2}}$ ${\displaystyle {\mathcal {D}}}$

Categories of engravings

A very important application is the category of engravings on a small category . Here is the category of quantities and you bet ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}={\mathcal {Set}}}$

${\displaystyle {\hat {\mathcal {C}}}:={\mathcal {Set}}^{{\mathcal {C}}^{op}}}$.

This is the functor category of the functors of the category to be dual in the category of sets. Such functors are called prawns . The Hom functors are examples and the assignment is called the Yoneda embedding of in . ${\displaystyle {\mathcal {C}}^{op}\rightarrow {\mathcal {Set}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {Hom} (-,C):{\mathcal {C}}^{op}\rightarrow {\mathcal {Set}}}$ ${\displaystyle C\mapsto \mathrm {Hom} (-,C)}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\hat {\mathcal {C}}}}$

Individual evidence

1. ^ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, definition 11.3
2. ^ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, definition 13.8
3. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag (2016), ISBN 978-3-662-53520-2 , definition 3.5.5
4. ↑ Martin Brandenburg: Introduction to Category Theory , Springer-Verlag (2016), ISBN 978-3-662-53520-2 , example 3.5.6
5. ↑ Saunders Mac Lane , Ieke Moerdijk : Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , chap. I: Categories of Functors