Under which conditions can we say that the cotensor objects in a (closed) V-category are the exponential objects? It is just when V=Set?
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No, this happens more often. Let me write $A\odot X$ and $X^A$ for tensors and cotensors in a category enriched over $\mathcal{V}$. I think you are (or might be) asking for a (natural) isomorphism between $A\odot X$ and $X^A$. In many algebraic situations, it is sensible to write $A\otimes X$ and $Hom(A,X)$. Writing $DA = Hom(A,k)$ in $\mathcal V$, where $k$ is the unit object of $\mathcal V$, it is then sometimes true that $Hom(A,X)$ is isomorphic to $DA\otimes X$. It is also sometimes true that $A$ is isomorphic to $DA$. That is true more often than it is true naturally, but if $\mathcal V$ is finite dimensional inner product spaces over the reals, for example, then there is a kind of contravariant naturality. This is not a complete answer of course, but it should give you a way to think about examples.
answered Jun 20, 2012 at 11:16
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$\begingroup$ Thanks. Did you mean perhaps Hom(A,X) iso DAtensorX? $\endgroup$ Commented Jun 21, 2012 at 11:00
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$\begingroup$ Are your "many algebraic situations" just abelian categories or do you have experience of cotensors in other settings? I am thinking of the category of algebras for some strong monad over $\mathcal V$. $\endgroup$ Commented Nov 28, 2015 at 14:06
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$\begingroup$ I rarely deal with abelian categories. I'm thinking for example of self-dual objects in symmetric monoidal categories, such as orbit $G$-spectra for a finite group $G$. Fausk, Hu, and I wrote a general categorical study of situations where left and right adjoints are isomorphic: math.uchicago.edu/~may/PAPERS/FormalFinalMarch.pdf $\endgroup$ Commented Nov 28, 2015 at 18:23