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Max New (Jun 08 2022 at 20:44):I have a situation where I have a functor $F : C \to D$ and I have subcategories $i: C_0 \hookrightarrow C$ and $j: D_0 \hookrightarrow D$ and I want to say something like $F$ has a right adjoint $G$, but the domain of $G$ is restricted to $D_0$, the codomain is restricted to $C_0$, but I want the UMP to hold in all of C, so I want
Is there a common name for this situation? The closest I could find is a $j$-relative right adjoint to $F$, but that doesn't require that the adjoint land in my subcategory $C_0$. I'd imagine this kind of situation comes up when you need to distinguish between large and small objects maybe, where you want to say some adjoint is only defined on small things and produces small things (not quite my scenario but similar in spirit).
John Baez (Jun 08 2022 at 22:07):HI! You need double dollars around all TeX here, not just displayed equations.
Nathanael Arkor (Jun 09 2022 at 02:36):Max New said:
I have a situation where I have a functor $F : C \to D$ and I have subcategories $i: C_0 \hookrightarrow C$ and $j: D_0 \hookrightarrow D$ and I want to say something like $F$ has a right adjoint $G$, but the domain of $G$ is restricted to $D_0$, the codomain is restricted to $C_0$, but I want the UMP to hold in all of C, so I want
Is there a common name for this situation? The closest I could find is a $j$-relative right adjoint to $F$, but that doesn't require that the adjoint land in my subcategory $C_0$. I'd imagine this kind of situation comes up when you need to distinguish between large and small objects maybe, where you want to say some adjoint is only defined on small things and produces small things (not quite my scenario but similar in spirit).
If I understand correctly, you're asking for . (This is often called a -relative adjunction in the literature, but I prefer the term "-relative coadjunction" to distinguish it from the dual notion since, unlike adjunctions, relative adjunctions are not self-dual; relative adjunctions induce relative monads, whilst relative coadjunctions induce relative comonads.)