I am looking for the correct terminology and a reference to the following construction.

Let $F:\mathcal C\to\mathcal D$ be a functor. Consider "cones" $(G,\alpha)$ where $G:\mathcal C\to\mathcal D$ is a functor such that all morphisms $G(f)$, $f\in Mor(\mathcal C)$, are invertible (i.e. $G$ is a functor to the core of $\mathcal D$), and $\alpha: G\Rightarrow F$ is a natural transformation. The "core limit" of $F$ is the universal "cone" $G_u$.

The "core limit" $G_u$ defines a "core limit object" $\hbox{core-lim } F=G_u(c)$ and a "monodromy group" $Mon(F)=G_u(\hom(c,c))$, $c\in\mathcal C$, which are unique up to isomorphism when $\mathcal C$ is connected. In this case, the real limit $\lim F$ is the invariant of $\hbox{core-lim } F$ by the action of $Mon(F)$.


1 Answer 1


I don't know if there is a standard terminology for this specific construction, but let me nonetheless say something. Let $|C|$ denote the groupoidification of $C$, i.e. the initial groupoid with a map $p:C\to |C|$.

Then what you are describing is the right Kan extension of $F: C\to D$ along $p: C\to |C|$. This recovers what you said at the end about the invariants of $G$ being exactly the limit of $F$, because right Kan extensions compose, and $\lim_C F$ is the right Kan extension of $F$ along $C\to *$, which you can right Kan extend in two steps: first to $|C|$, and second from $|C|$ to $*$ - this second step takes invariants for $G$ when $|C| = BG$ for some group $G$, i.e. when $C$ is (weakly) connected, but in general takes $\prod_{x\in\pi_0|C|} (\mathrm{core-lim})^{Aut(x)}$.

I haven't really seen this construction considered before (other than "it's a right Kan extension"), but maybe someone else can chime in and give references/terminology.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.