Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must also admit a dual. Can we describe the evaluation and coevaluation maps of $Y$ in terms of $\sigma, c$, and $e$?
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2 Answers
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This won't fit in a comment, maybe it's useful to someone:
$$ \begin{align*} (\mathrm{id}_Y⊗\mathrm{ev}_Y)∘(\mathrm{coev}_Y⊗\mathrm{id}_Y) &= (\mathrm{id}_Y⊗\mathrm{ev}_X)∘(\mathrm{id}_{Y⊗X^*}⊗σ^{-1}) ∘(σ⊗\mathrm{id}_{X^*⊗Y})∘(\mathrm{coev}_X⊗\mathrm{id}_Y) \\ &= (\mathrm{id}_Y⊗\mathrm{ev}_X)∘(σ⊗\mathrm{id}_{X^*}⊗σ^{-1})∘(\mathrm{coev}_X⊗\mathrm{id}_Y) \\ &= (σ⊗\mathrm{id}_1)∘(\mathrm{id}_X⊗\mathrm{ev}_X)∘(\mathrm{coev}_X⊗\mathrm{id}_X)∘(\mathrm{id}_1⊗σ^{-1}) \\ &= σ∘σ^{-1} \end{align*}$$
The point is that the unitors are usually suppressed in the triangle identities.
answered Apr 2 at 19:04
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$\begingroup$ Thanks a lot for spelling this out for me! Just a few things: At the end of the first line, I guess this should be $\mathrm{coev}_X \otimes \mathrm{id}_Y$? On the second line, the range of $\mathrm{coev} \otimes \mathrm{id}_Y$ is $X \otimes X^* \otimes Y$. How does $\sigma \otimes \sigma^{-1}$ act on $X \otimes X^* \otimes Y$? I guess as $\sigma \otimes \mathrm{id}_{X^*} \otimes \sigma^{-1}$ $\endgroup$ Commented Apr 2 at 20:18
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1$\begingroup$ @Yilmaz You might need to scroll to see the whole line. I did forget the middle id, thanks! Fixed now. $\endgroup$ Commented Apr 2 at 21:00
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$\begingroup$ Scrolling, yes, now I see :) Thanks a lot for this, it was driving me crazy for the last few days, now it is clear! $\endgroup$ Commented Apr 2 at 21:21
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Yes.
First, set $Y^* = X^*$.
Then, compose the evaluation and coevaluation maps of $X$ with maps like $\operatorname{id_{X^*}} \otimes \sigma$.
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1$\begingroup$ If I write $e_{\sigma} := e \circ (id \otimes \sigma^{-1})$ and $c_{\sigma} := (\sigma \otimes \mathrm{id}) \circ c$, then the relation $(id \otimes e_{\sigma}) \circ (c_{\sigma} \otimes id) = id$ does not seem to be satisfied, since the $\sigma$ and $\sigma^{-1}$ are in the wrong place for cancellation. Hence $e_{\sigma}$ and $c_{\sigma}$ do not satisfy the axioms of dual object. $\endgroup$ Commented Apr 2 at 10:55
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$\begingroup$ However for vector spaces and projective modules it seems to work . . . so I guess it works in general for reasons that I cannot see . . . $\endgroup$ Commented Apr 2 at 13:11