This is a notation question. Let $\mathbf{Set}$ denote the category of sets. What is the category $\mathbf{Set}^A$, where $A$ is a set?
I see it mentioned often, but without being defined. Is it the overcategory $A/\mathbf{Set}$?
What is the category $\mathbf{Set}^A$? (for a set $A$)
1
$\begingroup$
category-theory
-
0It is the $A$-fold product of $\text{Set}$ with itself, or said another way the category of functors $A \to \text{Set}$, thinking of $A$ as a discrete category (no non-identity morphisms). – 2017-01-22
-
0@QiaochuYuan Thanks! But, categorically, how is it constructed in the category of locally small categories? Is it some kind of power in the sense of enriched category theory? – 2017-01-22
-
0In any category it makes sense to talk about $A$-fold products where $A$ is a set. Writing down this definition if you haven't seen it before is a nice exercise. – 2017-01-22
-
0@QiaochuYuan Thanks again. I think I find the power/cotensor point of view more intuitive... – 2017-01-22
-
0Well, if a category has all $A$-fold products for every set $A$, that's equivalent to it being powered over $\text{Set}$. Similarly to having all $A$-fold coproducts and being copowered. Locally small categories is even powered and copowered over small categories. – 2017-01-22
-
0Very helpful. I will write down the details. – 2017-01-22
2 Answers
1
It is likely the functor category whose objects consist of the functors $A\to Set$ where $A$ is regarded as a discrete category (does that make sense based on your context? This is the same as $|A|$-tuples of sets whose morphisms are $|A|$-tuples of functions. In general, $C^D$ refers to the functor category between $D$ and $C$
-
4You have the notation backwards. $C^D$ means $D \to C$. Similarly, you mean $A \to \mathbf{Set}$ not $\mathbf{Set}\to A$. – 2017-01-22
1
$D^C$ is the category of functors $C \to D$ and natural transformations between them.
In $\mathbf{Set}^A$, $A$ is being regarded as a discrete category; $A$ is the set of objects, and the only arrows are identities.
There is a canonical equivalence $\mathbf{Set}/A \equiv \mathbf{Set}^A$. The forward direction associates to an object $B \xrightarrow{f} A$ of $\mathbf{Set}/A$ the functor $a \mapsto f^{-1}(a)$. The reverse direction associates to a functor $F : A \to \mathbf{Set}$ the object given by projection map of the disjoint union $\coprod_{a \in A} F(a) \to A$. (hopefully the rest of the details are obvious)
There is also the notion of an "internal category"; any object $X$ of a category $\mathcal{C}$ can be viewed internally as a discrete category $\mathcal{X}$. Inspired by the above, we actually define $\mathcal{C}^\mathcal{X} = \mathcal{C} / X$.