Terminology question: "core limit"

Question

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 $Mon(F)$.

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.