3
$\begingroup$

I'm having trouble in unserstanding a hom-functor.

Suppose we have a hom-functor $Hom(X,$_$)$ for some Category $\mathcal{C}$.

Suppose further that the hom-sets Hom$_{Set}(X,A)$ and Hom$_{Set}(X,B)$ are empty and there exists an arrow $f$ going from $A$ to $B$.

What would be the lifted function $\mathcal{C}(X,f)$ in category $Set$?

Isnt't this a function going from the empty set to the empty set?

How does this work?

$\endgroup$
6
$\begingroup$

First, note that the notation for the hom-set is $\mathrm{Hom}_{\mathcal C}(X, A)$, rather than $\mathrm{Hom}_{\mathbf{Set}}(X, A)$ (the $\mathbf{Set}$ is implicit).

Yes, $\mathrm{Hom}_{\mathcal C}(X, f)$ will be a function $\emptyset \to \emptyset$ if $\mathrm{Hom}_{\mathcal C}(X, A)$ and $\mathrm{Hom}_{\mathcal C}(X, B)$ are empty. But that's okay! There is exactly one function from the empty set to the empty set, which is the identity function. (In fact, for each set $Y$, there's exactly one function from the empty set to $Y$. We just have to provide an assignment of each element in the domain to an element of the codomain: when the domain is empty, this is trivial, as there aren't any elements to provide assingments for.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.