Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

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 $DA\otimes V$. 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.

share|improve this answer
Thanks. Did you mean perhaps Hom(A,X) iso DAtensorX? –  Doctor Gibarian Jun 21 '12 at 11:00
Yes, of course. –  Peter May Jun 25 '12 at 12:52
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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