Why does this collection is a set and what is this symbol?

Question

My question is quite simple, I'm trying to understand why $Hom_{sets}(X,Y)$ is a set and what is this symbol $Y^X$?

Definition 1.12. A category $\mathbf{C}$ is called locally small if for all objects $X, Y$ in $\mathbf{C}$, the collection $\textrm{Hom}_{\mathbf{C}}(X,Y) = \{f \in \mathbf{C}_1 \mid f : X \longrightarrow Y \}$ is a set (called a hom-set).

Many of the large categories we want to consider are in fact locally small. $\mathbf{Sets}$ is locally small since $\color{purple}{\textrm{Hom}_{\mathbf{Sets}}(X,Y) = Y^X}$, the set of all functions from $X$ to $Y$. Similarly, $\mathbf{Pos}$, $\mathbf{Top}$, and $\mathbf{Group}$ are all locally small (is $\mathbf{Cat}$?), and, of course, any small category is locally small.

I'm sorry, I'm sure this should be a silly question, but I only know very basic set theory.

Thanks in advance.

Answer 1

Note that if $X,Y$ are fixed then we can regard a function as its graph, rather than an ordered triplet (as the praxis of category theory goes).

This graph is always a subset of $X\times Y$.

Therefore the collection $\{f\subseteq X\times Y\mid f\text{ is a function from }X\text{ to }Y\}$ is a subset of $\mathcal P(X\times Y)$.

So we have that:

1. $X\times Y$ is a set. It has a power set.
2. Therefore $\mathcal P(X\times Y)$ is a set. Its definable collections are sets.
3. Therefore $\{f\subseteq X\times Y\mid f\text{ is a function from }X\text{ to }Y\}$ is a set. We denote this set as $Y^X$ in set theory.

So if $X$ and $Y$ are sets the above shows that $Y^X=\hom_{\bf Sets}(X,Y)$ is also a set; and thus $\bf Sets$ is locally small.