PlanetMath: functor category

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (21)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

functor category

(Definition)

Let $\mathcal{C},\mathcal{D}$ be categories. Consider the class of all covariant functors $T:\mathcal{C}\to \mathcal{D}$ , and the class of all natural transformations $\tau:S\dot{\to} T$ for every pair $S,T:\mathcal{C}\to \mathcal{D}$ of functors. Write $\mathcal{D}^\mathcal{C}$ for the pair .

For each pair of functors $S,T:\mathcal{C}\to \mathcal{D}$ , write $\hom(S,T)$ the class of all natural transformations from to . If $\tau$ is in both $\hom(S,T)$ and $\hom(U,V)$ , then and .

Using the composition of natural transformations, we have a mapping

$\displaystyle \bullet:\hom(R,S)\times \hom(S,T)\to \hom(R,T),$

for every triple $R,S,T:\mathcal{C}\to \mathcal{D}$ . Since composition of natural transformations is associative, the associativity of $\bullet$ applies.

In addition, for each $S:\mathcal{C}\to \mathcal{D}$ , we have the identity natural transformation $1_S\in \hom(S,S)$ . For every $\tau\in \hom(S,T)$ and every $\eta\in \hom(T,S)$ , we have $\tau \bullet 1_S=\tau$ and $1_S\bullet \eta=\eta$ .

From the discussion above, we are ready to call $\mathcal{D}^\mathcal{C}$ a category. However, unless $\hom(S,T)$ is a set for every pair of functors in , $\mathcal{D}^\mathcal{C}$ is not a category. When $\mathcal{D}^\mathcal{C}$ is a category, we call it the category of functors from $\mathcal{C}$ to $\mathcal{D}$ , or simply a functor category.

That $\mathcal{D}^\mathcal{C}$ is a functor category depends on various restrictions being placed on the “sizes” of $\mathcal{C}$ and $\mathcal{D}$ :

Proposition 1 If $\mathcal{C}$ is $\mathcal{U}$ -small, then $\mathcal{D}^\mathcal{C}$ is a category.

Proof. Suppose $\mathcal{C}$ is $\mathcal{U}$ -small. Consider the class $\hom(S,T)$ . Each $\tau \in \hom(S,T)$ is determined by the collection of morphisms $S(A)\to T(A)$ for each object

in $\mathcal{C}$ . This means that, for each

in $\mathcal{C}$ , $\hom(S(A),T(A))$ contains the image of every $\tau\in \hom(S,T)$ under

. So the class of all these natural transformations is a subclass of the product

$\displaystyle \prod_{A\in \operatorname{Ob}(\mathcal{C})} \hom(S(A),T(A))$

(1)

Since $\operatorname{Ob}(\mathcal{C})$ , as well as each $\hom(S(A),T(A))$ is a set, so is the product (1). Hence $\hom(S,T)$ , being a subclass of (1), is a set, or that $\mathcal{D}^{\mathcal{C}}$ is a category. $\qedsymbol$

Proposition 2 If in addition $\mathcal{D}$ is a $\mathcal{U}$ -category, then so is $\mathcal{D}^\mathcal{C}$ .

Proof. $\mathcal{D}$ being a $\mathcal{U}$ -category means that $\hom(S(A),T(A))$ is $\mathcal{U}$ -small, for every object

in $\mathcal{C}$ . Since $\operatorname{Ob}(\mathcal{C})$ is also $\mathcal{U}$ -small (assumption in Proposition 1), the product (1) above is $\mathcal{U}$ -small. Consequently, $\hom(S,T)$ , being a subclass of (1), is $\mathcal{U}$ -small. This shows that $\mathcal{D}^{\mathcal{C}}$ is a $\mathcal{U}$ -category. $\qedsymbol$

Proposition 3 If $\mathcal{D}$ is furthermore $\mathcal{U}$ -small, so is $\mathcal{D}^\mathcal{C}$ .

Proof. We want to show that the class $\mathcal{M}$ of all functors from $\mathcal{C}$ to $\mathcal{D}$ is $\mathcal{U}$ -small. A functor $S:\mathcal{C}\to \mathcal{D}$ can be broken up into two components: a function $S_1: \operatorname{Ob}(\mathcal{C})\to \operatorname{Ob}(\mathcal{D})$ , and a function $S_2:\operatorname{Mor}(\mathcal{C})\to \operatorname{Mor}(\mathcal{D})$ , so that $S_2(A\to B)=S_1(A)\to S_1(B)$ .

Define a binary relation $\sim$ on $\mathcal{M}$ so that $S\sim T$ iff they have the same first component: . It is easy to see that $\sim$ is an equivalence relation on $\mathcal{M}$ . Let be the equivalence class containing the functor . For every morphism $A\to B$ , its image under the second component of every functor in lies in $\hom(S_1(A),S_1(B))$ . So the size of can not exceed the size of

$\displaystyle \prod_{A,B\in \operatorname{Ob}(\mathcal{C})} \hom(S_1(A),S_1(B))$

Since $\operatorname{Ob}(\mathcal{C})$ is $\mathcal{U}$ -small (assumption in Prop 1), so is $\operatorname{Ob}(\mathcal{C})\times \operatorname{Ob}(\mathcal{C})$ . Furthermore, since each $\hom(S_1(A),S_1(B))$ is $\mathcal{U}$ -small (assumption in Prop 2),

is $\mathcal{U}$ -small as well.

Next, let us estimate the size of the class $\mathcal{M}/\sim$ of equivalence classes in $\mathcal{M}$ . First, note that for every functor $S:\mathcal{C}\to \mathcal{D}$ , its first component is a function from the set $\operatorname{Ob}(\mathcal{C})$ to the set $\operatorname{Ob}(\mathcal{D})$ by assumption. As $[S]\ne [T]$ iff $S_1\ne T_1$ , the size can not exceed

$\displaystyle \vert\operatorname{Ob}(\mathcal{D})^{\operatorname{Ob}(\mathcal{C})}\vert$

the cardinality of the set of all functions from $\operatorname{Ob}(\mathcal{C})$ to $\operatorname{Ob}(\mathcal{D})$ . By assumption, $\operatorname{Ob}(\mathcal{D})$ is $\mathcal{U}$ -small, so is $\operatorname{Ob}(\mathcal{D})^{\operatorname{Ob}(\mathcal{C})}$ . As a result, $\mathcal{M}/\sim$ is $\mathcal{U}$ -small. Together with the fact that

is $\mathcal{U}$ -small for each functor

, we have that $\mathcal{M}$ itself must be $\mathcal{U}$ -small, which completes the proof. $\qedsymbol$

"functor category" is owned by CWoo.

(view preamble | get metadata)

Also defines:

category of functors

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: proof, completes, estimate, size, equivalence class, equivalence relation, easy to see, iff, binary relation, function, components, proposition, product, subclass, image, contains, object, morphisms, collection, restrictions, identity natural transformation, addition, associative, mapping, composition of natural transformations, natural transformations, covariant functors, class, categories
There are 7 references to this entry.

This is version 5 of functor category, born on 2008-09-23, modified 2008-09-30.
Object id is 11082, canonical name is FunctorCategory.
Accessed 250 times total.

Classification:

AMS MSC:	18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
	18A25 (Category theory; homological algebra :: General theory of categories and functors :: Functor categories, comma categories)
	18-00 (Category theory; homological algebra :: General reference works )

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact