Suppose $\mathcal{E}$ is an elementary topos (take as a definition that of Mac Lane and Moerdijk "Sheaves in Geometry and Logic"). I have a problem with a fact concerning the cartesian closure of $\mathcal{E}$: suppose we have two arrows $f:A \to B, g:B \to C$. By considering $1\times A \simeq A, 1\times B \simeq B$ we can transpose these two arrows obtaining $\hat f:1 \to B^A, \ \hat g:1 \to C^B$.

What I want to show is that $ev_B \circ (\hat g \times f)=g \circ f \circ \pi_A:1 \times A \to C$, where $ev_B:C^B \times B \to C$ is the $B$-component of the counit of the adjunction $-\times B \dashv (-)^B$. This it obviously true for $\mathcal{E}:= Sets$, but I can't find the right diagram to prove it in full generality (obviously it will be true for any cartesian closed category).

Thanks in advance!

  • 1
    $\begingroup$ Mac Lane and Moerdijk give several definitions in the book, at least one of which does not assume cartesian closedness. Your question is solely about cartesian closed categories, so you need not refer to toposes at all. $\endgroup$ – Zhen Lin Oct 26 '13 at 12:57
  • $\begingroup$ Yeah, but that's a useless puntualization since I've already written it! $\endgroup$ – Edoardo Lanari Oct 26 '13 at 13:20

Recall that there is a hom-set bijection $$\mathrm{Hom}(X \times Y, Z) \cong \mathrm{Hom}(X, Z^Y)$$ that is natural in $X$, $Y$, and $Z$. That implies there is a natural morphism $\mathrm{ev}_{Y,Z} : Z^Y \times Y \to Z$ inducing this bijection, with the right-to-left bijection being $\hat{g} \mapsto \mathrm{ev}_{Y,Z} \circ (\hat{g} \times \mathrm{id}_Y)$. In particular, $$\mathrm{ev}_{Y,Z} \circ (\hat{g} \times f) = \mathrm{ev}_{Y,Z} \circ (\hat{g} \times \mathrm{id}_Y) \circ (\mathrm{id}_X \times f)= g \circ (\mathrm{id}_X \times f)$$ and taking $X = 1$ and using the naturality of the canonical isomorphism $1 \times T \cong T$, we may deduce the required equation.

  • $\begingroup$ Thank you very much! Indeed it was easy, I only needed to factorize that map! $\endgroup$ – Edoardo Lanari Oct 26 '13 at 13:40

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