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Stream: theory: category theory

Topic: Stuck trying to recognize the pattern

Patrick Nicodemus (May 27 2025 at 16:26):

I'm looking at a situation and trying to figure out the right pattern to describe it. Reminds me of a relative left adjoint but I'm stuck.

Theorem: Let $A,B, C$ be categories, with $H : C\to B, G : B \to A$. Suppose that $G\circ H$ has a left adjoint, $F$. Suppose that $Hom(b, H(c))\cong Hom(G(b), G(H(c))$ naturally in $b,c$. Then $F\circ G$ is left adjoint to $H$.

Proof: $Hom(F(G(b),c) \cong Hom(G(b), G(H(c)) \cong Hom(b, H(c))$.

Simple proof but I just don't recognize the condition $Hom(b, H(c))\cong Hom(G(b), G(H(c))$ as having a common name. It's like $G$ is "faithful relative to $H$" or something.

Patrick Nicodemus (May 27 2025 at 16:38):

Example: let $X$ be a topological space and $\mathcal{B}$ a basis. Let $u_X : Sh(X) \to Psh(X)$ be the forgetful functor from sheaves to presheaves, and $i_\mathcal{B} : Psh(X)\to Psh(\mathcal{B})$ the forgetful functor from presheaves on all open sets to presheaves on basis open sets.

Then $Nat(\mathcal{F},u_X(\mathcal{G}))\cong Nat(i_\mathcal{B}(\mathcal{F}), i_\mathcal{B}(u_X(\mathcal{G}))$, naturally in $\mathcal{F}$ and $\mathcal{G}$.

Nathanael Arkor (May 27 2025 at 18:41):

The condition states that $G$ is left $GH$-coadjoint to $H$: it's an example of a [[relative coadjunction]] (unlike for non-relative adjunctions, where adjunctions are indistinguishable from coadjoints, it's necessary to distinguish between relative adjunctions and relative coadjunctions).

Nathanael Arkor (May 27 2025 at 18:46):

In my thesis, I called the situation where you have a relative adjunction of the form $L \vdash_{RL} R$ a resolute pair (because they an important property with respect to resolutions of (relative) monads; see Proposition 6.1.6). The situation you are considering is the dual; it might be called a "co-resolute pair" (though this should be viewed as referring to resolutions of (relative) comonads, rather than "co-resolutions").

Nathanael Arkor (May 27 2025 at 18:47):

See also Corollary 5.32 of The formal theory of relative monads.