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I'm currently working through David Spivak's Category Theory for Scientists, and I'd just like to verify that I am understanding $\def\homset{\operatorname{Hom}_{\mathsf{Set}}}\homset(X,Y)$ correctly. My (informal) understanding that is that it denotes the set of all the different functions from $X \rightarrow Y$. Thus, if we let $A = \{1,2,3,4,5\}$ and $B = \{x,y\}$, we have the following answers to these questions:

a) How many elements does $\homset(A,B)$ have?

  • 32 since each element in $A$ can map to one of two elements in $B$. Thus, we have $2^5 = 32$.

b) Find a set A such that for all sets $X$ there is exactly one element in $\homset(X, A)$.

  • If there is exactly on element in the hom-set, this means we can only have one function from $X$ to $Y$. Thus, $A$ can be any set containing only one element.

c) Find a set $B$ such that for all sets $X$ there is exactly one element in $\homset(B, X)$.

  • This is the one that I'm stuck on and made me think that perhaps I'm misunderstanding the definition given, because by what I'm given, such a set can't exist.

Could someone please confirm my thinking or clarify what I might be misunderstanding? Thanks!

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Don't forget your friend the empty set! – Trevor Wilson Oct 2 '13 at 4:54
I was actually thinking about that. Wasn't sure that a function could map the empty set to something, though I suppose on hind thought that the definition definitely would allow it. Thanks! – pomegranate Oct 2 '13 at 5:08
@promegranate The function (you wasn't sure about) from $\emptyset$ to any $X$ is just the injection $\emptyset \subseteq X$. – Pece Oct 2 '13 at 10:10
up vote 4 down vote accepted

It turns out that my reasoning was okay; in (c), the empty set could be such a set. Thanks to Trevor Wilson for the reminder!

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A nice analogy for you:

Maps from B to X can be represented in category theory as $X^{B}$.

If B is the empty set, with no elements, this is analogous to putting X to the 0th power, which equals 1 in elementary mathematics. 1 element in the set of maps.

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