realisation-and-nerve adjunction


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

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

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


Let Δ C:ΔC\Delta_C:\Delta\to C be a cosimplicial object of CC.

N(A) n:=C(Δ C[n],A) N(A)_n :=C(\Delta_C[n],A)
(|| FN F)(||_F\dashv N_F)

Last revised on November 12, 2012 at 02:20:52. See the history of this page for a list of all contributions to it.