The Unapologetic Mathematician

Mathematics for the interested outsider

Enriched Adjunctions

When I started in on adjoint functors, I gave the definition in terms of a bijection of hom-sets. Then I showed that we can also specify it in terms of its unit and counit. Both approaches (and their relationship) generalize to the enriched setting.

Given a functor F:\mathcal{C}\rightarrow\mathcal{D} and another G:\mathcal{D}\rightarrow\mathcal{C}, an adjunction is given by natural transformations \eta:1_\mathcal{C}\rightarrow G\circ F and \epsilon:F\circ G\rightarrow1_\mathcal{D}. These transformations must satisfy the equations (1_G\circ\epsilon)\cdot(\eta\circ1_G)=1_G and (\epsilon\circ1_F)\cdot(1_F\circ\eta)=1_F. By the weak Yoneda Lemma, this is equivalent to giving a \mathcal{V}-natural isomorphism \phi_{C,D}:\hom_\mathcal{D}(F(C),D)\rightarrow\hom_\mathcal{C}(C,G(D)).

Indeed, a \mathcal{V}-natural transformation in this direction must be of the form \phi_{C,D}=\hom_\mathcal{C}(\eta_C,1_{G(D)})\circ G_{F(C),D}, and one in the other direction must be of the form \varphi_{C,D}=\hom_\mathcal{D}(1_{F(C)},\epsilon_D)\circ F_{C,G(D)}. The equations \phi\circ\varphi=1 and \varphi\circ\phi=1 are equivalent, by the weak Yoneda Lemma, to the equations satisfied by the unit and counit of an adjunction.

The 2-functor \mathcal{V}\mathbf{-Cat}\rightarrow\mathbf{Cat} that sends an enriched category to its underlying ordinary category sends an enriched adjunction to an ordinary adjunction. The function underlying the \mathcal{V}-natural isomorphism \phi is the bijection of this underlying adjunction.

As we saw before, a \mathcal{V}-functor G has a left adjoint F if and only if \hom_\mathcal{D}(D,F(\underline{\hphantom{X}})) is representable for each D\in\mathcal{D}. Also, an enriched equivalence is an enriched adjunction whose unit and counit are both \mathcal{V}-natural isomorphisms. Just as for ordinary adjunctions, we have transformations between enriched adjunctions, a category of enriched adjunctions between two enriched categories, enriched adjunctions with parameters, and so on.

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September 4, 2007 - Posted by | Category theory

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