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Stream: theory: category theory

Topic: relative left adjoints, categories of fractions

Patrick Nicodemus (Apr 21 2022 at 06:00):

Let $\mathcal{C}$, $\mathcal{D}$ be categories, $F : \mathcal{C}\to \mathcal{D}$, $G : \mathcal{D}\to \mathcal{C}$, $F\dashv G$.

If $f : A \to B$ is a morphism in $\mathcal{D}$, and $U(f)$ is an isomorphism, then for each object $X$ in $\mathcal{C}$, and each morphism $g : F(X)\to B$, there exists a unique lift of $g$ along $f$.

The converse is also true. If $f : A\to B$ has the property that for each $X$ in $\mathcal{C}$, the induced map $\operatorname{Hom}(F(X),f) : \operatorname{Hom}(F(X),A) \to \operatorname{Hom}(F(X),B)$ is a bijection, then $U(f)$ is an isomorphism.

The class $\Sigma \subset \operatorname{Mor}(\mathcal{D})$ of morphisms $f$ such that $U(f)$ is an isomorphism is important in the study of categories of fractions, as if $F$ is fully faithful and $\mathcal{C}$ is a coreflective subcategory of $\mathcal{D}$, then $\mathcal{C}$ is equivalent to the localization of $\mathcal{D}$ at the morphisms of $\Sigma$.

I have a situation where I have a (fully faithful) functor going in one direction, $F: \mathcal{C}\to \mathcal{D}$, but I haven't been able to prove the existence of a globally defined right adjoint without further hypotheses. Instead I have a partially defined right adjoint, i.e. a relative right adjoint; I have a full subcategory $\mathcal{X}$ of $\mathcal{D}$ and I have a relative right adjoint to $F$ along the inclusion functor $i_{\mathcal{X}}$, call this right adjoint $B$.

I'm interested in the kind of unique lifting property I mentioned before, but it's more awkward in this situation because the adjoint is only partial. I'm thinking about maps $m : D\to i_{\mathcal{X}}(X)$ with the property that for all $C \in \operatorname{Obj}(\mathcal{C})$, the induced map $\operatorname{Hom}(F(C),D)\to \operatorname{Hom}(F(C),i_{\mathcal{X}}(X))$ is a bijection. Thus $\operatorname{Hom}(F(C),D)\cong \operatorname{Hom}(C,B(X))$.

Do these kinds of morphisms have a name? Is anything known about them, or is there any way to characterize them?
It's an awkward concept but things are bound to get awkward when you have three categories floating around.