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Stream: theory: category theory

Topic: a variant of adjunctions

Mike Shulman (Jun 19 2021 at 17:04):

In a monoidal 2-category $(\mathcal{K},\otimes,I)$, suppose I have morphisms $f:I\to A\otimes B$ and $g:B\otimes A \to I$ with 2-cells $\eta : 1_A \to (1_A \otimes g)(f\otimes 1_A)$ and $\epsilon : (g\otimes 1_B)(1_B\otimes f) \to 1_B$, satisfying two evident "triangle" identities. Has this situation been studied? Does it have a name?

Mike Shulman (Jun 19 2021 at 17:08):

Motivation: if $\mathcal{K}$ is compact closed with dual objects $A^o$, this is a way to re-express an ordinary adjunction $A^o \rightleftarrows B$ without reference to duals. In particular, a contravariant functor $h:A^{\mathrm{op}} \to B$ in $\rm Cat$ induces a structure of this sort in $\rm Prof$, where $f(a,b) = B(b,h a)$ and $g(b,a) = B(h a,b)$.

Amar Hadzihasanovic (Jun 19 2021 at 18:13):

If you deloop the monoidal 2-category to a tricategory, this seems to be the same data as what is called a "lax 2-adjunction" between $A$ and $B$ here on the nLab.

Mike Shulman (Jun 19 2021 at 23:51):

That's true. So I suppose it could be called a "lax 2-duality". Has anyone studied it before?