# Generalization of topological realization of simplicial sets to general categories.

Given a functor $$U: \Delta \rightarrow \textbf C$$ from the simplex category into $$\textbf C$$, I want to show that if $$\mid -\mid$$ is "geometric realization" of a simplicial set in $$\textbf C$$ given by the left Kan extension (assuming it exists)

the diagram commutes. I.e $$\mid \text{yoneda}(-)\mid = U(-)$$. This is true for the topological case ($$\textbf{C} = \text{Top})$$ but I don't know about the general case.

• The result quoted here (math.stackexchange.com/questions/3435524/…) is related. The idea is that in a nice case (if $C$ is cocomplete) $|-|$ is just an extension by colimits Dec 7, 2019 at 17:02

This is true in considerable generality. If $$y:A\to B$$ is fully faithful and $$f:A\to C$$, then $$Lan_u f\circ u$$ is naturally isomorphic to $$f$$ whenever $$Lan_u$$ is a pointwise left Kan extension, for instance, whenever $$C$$ is cocomplete. The reason is that $$Lan_u f(b)$$ is the colimit of the diagram indexed by the comma category $$u/b$$ whose value at $$u(a)\to b$$ is simply $$f(a)$$. When $$u$$ is fully faithful, $$u/u(a)$$ is equivalent to $$A/a$$, so has a terminal object $$id_a:u(a)\to u(a)$$, and the colimit is the value $$f(a)$$ taken at $$id_a$$.