What is the category $\mathbf{Set}^A$? (for a set $A$)

Question

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 $A/\mathbf{Set}$?

Answer 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 $C$

Answer 2

$D^C$ 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 $\mathcal{C}^\mathcal{X} = \mathcal{C} / X$.