Let $X^Y := \mathsf{Hom}_{\mathcal{AB}}(Y, X)$ be a set of all morphisms from objects $Y$ to $X$ from Abelian groups category $\mathcal{AB}$. Let $X \times Y$ be a product and $X + Y$ be a coproduct of object $X$ and $Y$.

The task is to establish canonical isomorphisms: $$Z^{Y \times X} \cong (Z^Y)^X, \quad Z^{Y+X} \cong Z^Y\times Z^X, \quad (Z \times Y)^X \cong Z^X \times Y^X.$$

I understand the task, but I dont know, how to do it specifically for $\mathcal{AB}$ (nor for any other category). It would be nice, if someone will provide me a proof please, or at least link to a source.

  • $\begingroup$ $X$, $Y$, $Z$ are abelian groups, right? $\endgroup$ – Oskar Apr 8 '16 at 22:36
  • 1
    $\begingroup$ The idea is to find a morphism $g: X \to Z^Y$ that sends $x \mapsto g_x$, where for $f \in Z^{X\times Y}$ we have $f(x,y) = g_x(y)$. Google "currying". $\endgroup$ – David Wheeler Apr 8 '16 at 23:30
  • 2
    $\begingroup$ The first isomorphism listed is false. $\endgroup$ – Zhen Lin Apr 9 '16 at 6:41
  • $\begingroup$ Instead, it holds as $Z^{Y\otimes X}\cong (Z^Y)^X$. $\endgroup$ – Berci Apr 11 '16 at 21:06

Your Answer

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

Browse other questions tagged or ask your own question.