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

Recall in a $2$-category $X$, a $1$-cell $f:X\to Y$ is called an equivalence provided there exists a $1$-cell $g:Y\to X$ together with the data of a pair of isomorphisms $\eta_X: gf \to \operatorname{id}_X$ and $\eta_Y: fg \to \operatorname{id}_Y$.

Then suppose we impose the additional requirement that $f\ast \eta_X = \eta_Y \ast f$ and $g\ast \eta_Y = \eta_X \ast g$.

Does this idea have a name? Is it the case that for any equivalence, we can find a pair $\eta_X, \eta_Y$ for which this holds?

share|improve this question

1 Answer 1

up vote 7 down vote accepted

This is well-known under the name adjoint equivalence (when you replace $\eta_Y$ by $\eta_Y^{-1}$ in your data). And yes, every equivalence can be modified to some adjoint equivalence (one may fix the unit, but has to modify the counit; or vice versa).

share|improve this answer

Your Answer

 
discard

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.