Spahn realisation-and-nerve adjunction

Definition

Let $F:C\to D$ be a $V$-enriched functor of $V$enriched categories, let $j:C\to [C^{op},V]$ be the $V$-enriched Yoneda embedding.

The V-enriched Yoneda extension? of $F$ - i.e. the left Kan extension? $\Lan_j F$ of $F$ along $j$ is also called realization functor associated to $F$ and in this context denoted by $||_F$.

The functor $N:\begin{cases}D^{op}\to V\\c\to j(c)\circ F^{op}\end{cases}$ is called nerve functor associated to $F$.

Example

Let $\Delta_C:\Delta\to C$ be a cosimplicial object of $C$.

$$N(A)_n :=C(\Delta_C[n],A)$$

Proposition

$$(||_F\dashv N_F)$$