# Are extensions of simplicial objects to functors $\mathsf{sSet} \to \mathsf{C}$ Kan extensions?

Suppose that we have a functor $$F : \boldsymbol{\Delta}^\bullet \to \mathsf{C}$$ with domain the full subcategory of simplicial sets given by representable functors. For example, for each $$\Delta^n = \hom(n,-)$$ we can assign to it its baricentric subdivision $$\mathsf{sd} \Delta^n \in \mathsf{sSet}$$, or its geometric realization $$|\Delta^n| \in \mathsf{Top}$$.

By the Yoneda embedding, we have a fully faithful injective on objects functor $$i: \Delta^{op} \hookrightarrow \boldsymbol{\Delta}$$, hence $$F$$ can be thought of as a simplicial object

$$F : \Delta^{op} \to \mathsf{C}.$$

On the other hand, if $$X$$ is any simplicial set, we know that it is a colimit of representables

$$X = \mathsf{colim}_{\Delta^n \to X} \Delta^n.$$

If $$\mathsf{C}$$ is cocomplete, the definition

$$\widetilde{F}X := \mathsf{colim}_{\Delta^n \to X} F\Delta^n, \tag{1}$$

makes sense and gives an extension of $$F$$ to a functor $$\widetilde{F} : \mathsf{sSet} \to \mathsf{C}$$.

In other terms, we are using that simplicial sets are the free cocompletion of $$\Delta$$, and so this is the universal cocontinuous extension of $$F$$.

If I am not mistaken, since $$Fk = F\Delta^k$$, using the cone leg arrows the maps

$$Fk \to F\Delta^k \hookrightarrow \mathsf{colim}_{\Delta^n \to X} F\Delta^k= \widetilde{F}\Delta^n$$

gives a natural transformation $$\eta : F\Rightarrow \widetilde{F}i$$. So, assuming the former is correct, my question is:

Is $$(\widetilde{F},\eta)$$ a left Kan extension of $$F$$ along $$i$$?

I would also be interested in knowing what happens when we consider right Kan extensions, if these coincide and if not, what other interesting extension constructions can be made.

• Yes, it is the left Kan extension. The same formula can be used to construct both. – Zhen Lin Aug 30 '20 at 13:07

The fact that every functor $$F$$ like yours, with cocomplete codomain, admits a (essentially unique ) extension to $$sSet$$ amounts to the universal property of the free cocompletion, yes; and yes, the extension (has a right adjoint, called the $$F$$-nerve) and is a left Kan extension, along the Yoneda embedding $$y : \Delta \to {\sf sSet}$$.
As for right extensions, that's another story: the opposite of the category of presheaves on $$\Delta^{op}$$, i.e. the category $$[\Delta, {\sf Set}]^{op}$$, exhibits the universal property of the free completion of $$\Delta$$, and the contravariant Yoneda embedding $$y^\sharp : \Delta^{op}\to [\Delta, {\sf Set}]$$ yield a continuous extension for every $$G$$ with complete domain.
Usually, even assuming $$\sf C$$ bicomplete, it is not the case that $$\text{Lan}_y F \cong \text{Ran}_{y^\sharp} F$$.