The Unapologetic Mathematician

Mathematics for the interested outsider

Now that we know how to transform adjoints, we can talk about whole families of adjoints parametrized by some other category. That is, for each object $P$ of the parametrizing category $\mathcal{P}$ we’ll have an adjunction $F_P\dashv G_P:\mathcal{C}\rightarrow\mathcal{D}$, and for each morphism of $\mathcal{P}$ we’ll have a transformation of the adjunctions.

Let’s actually approach this from a slightly different angle. Say we have a functor $F:\mathcal{C}\times\mathcal{P}\rightarrow\mathcal{D}$, and that for each $P\in\mathcal{P}$ the functor $F(\underline{\hphantom{X}},P):\mathcal{C}\rightarrow\mathcal{D}$ has a right adjoint $G(P,\underline{\hphantom{X}})$. Then I claim that there is a unique way to make $G$ into a functor from $\mathcal{P}^\mathrm{op}\times\mathcal{D}$ to $\mathcal{C}$ so that the bijection $\phi_{C,P,D}:\hom_\mathcal{D}(F(C,P),D)\rightarrow\hom_\mathcal{C}(C,G(P,D))$ is natural in all three variables. Note that $G$ must be contravariant in $P$ here to make the composite functors have the same variance in $P$.

If we hold $P$ fixed, the bijection is already natural in $C$ and $D$. Let’s hold $C$ and $D$ fixed and see how to make it natural in $P$. The components $\phi_P:\hom_\mathcal{D}(F(C,P),D)\rightarrow\hom_\mathcal{C}(C,G(P,D))$ are already given in the setup, so we can’t change them. What we need are functions $\hom_\mathcal{D}(F(1_C,p),1_D):\hom_\mathcal{D}(F(C,P'),D)\rightarrow\hom_\mathcal{D}(F(C,P),D)$ and $\hom_\mathcal{C}(1_C,G(p,1_D)):\hom_\mathcal{C}(C,G(P',D))\rightarrow\hom_\mathcal{C}(C,G(P,D))$ for each arrow $p:P\rightarrow P'$.

For naturality to hold, we need $\hom_\mathcal{C}(1_C,G(p,1_D))\circ\phi_{P'}=\phi_P\circ\hom_\mathcal{D}(F(1_C,p),1_D)$. But from what we saw last time this just means that the pair of natural transformations $(F(1_C,p),G(p,1_D))$ forms a conjugate pair from $F(\underline{\hphantom{X}},P)\dashv G(P,\underline{\hphantom{X}})$ to $F(\underline{\hphantom{X}},P')\dashv G(P',\underline{\hphantom{X}})$. And this lets us define $G(p,1_D)$ uniquely in terms of $F(1_C,p)$, the counit $\epsilon'$ of $F(\underline{\hphantom{X}},P')\dashv G(P',\underline{\hphantom{X}})$, and the unit $\eta$ of $F(\underline{\hphantom{X}},P)\dashv G(P,\underline{\hphantom{X}})$ by using the first of the four listed equalities.

From here, it’s straightforward to show that this definition of how $G$ acts on morphisms of $\mathcal{P}$ makes it functorial in both variables, proving the claim. We can also flip back to the original viewpoint to define an adjunction between categories $\mathcal{C}$ and $\mathcal{D}$ parametrized by the category $\mathcal{P}$ as a functor from $\mathcal{P}$ to the category $\mathbf{Adj}(\mathcal{C},\mathcal{D})$ of adjunctions between those two categories.

July 31, 2007 Posted by | Category theory | 4 Comments