Holmstrom Nerve

nLab

http://ncatlab.org/nlab/show/nerve+and+realization

http://mathoverflow.net/questions/68293/nerves-of-simplicial-objects-in-categories-waldhausens-s-construction

http://mathoverflow.net/questions/117401/is-every-functor-inducing-a-homotopy-equivalence-a-composition-of-adjoint-functor claims that any adjoint functor induced homotopy equiv on the nerve

For any (small?) category $C$, one defines a simplicial set $BC$, called its nerve. The $q$-simplices of a category $C$ is the set of functors $[q] \to C$, see Dundas, page 24, for details. This construction is an extension of the Yoneda embedding $\Delta \to Sset$, and embeds $Cat$ as a full subcategory of $Sset$. The nerve has a left adjoint.

If $C$ is given by a group $G$, then $BG$ is a $K(G,1)$ space.

Every nerve of a group is a Kan complex, but not every nerve of a category. Actually, a nerve of a small category is a fibrant simplicial set iff the category is a groupoid.

Remark: Every simplicial group (regarded as a simplicial set) is fibrant.

Cox: Homotopy limits and the homotopy type of functor categories; has an alternative description of the nerve of a small category. http://www.ams.org/mathscinet-getitem?mr=0407022

Ref: Page 10 of Baues: Homotopy types (e)

Nerves for bicats, see http://arxiv.org/abs/0903.5058

Q from Maltsiniotis to ALGTOP: Let $N$ be the nerve functor from categories to simplicial sets and $c$ its left adjoint. It is well known that $c$ (or equivalently $Nc$) is not compatible with weak equivalences, but what about weakly contractible objects? Grothendieck asks in Pursuing Stacks to find a weakly contractible simplicial set $X$ such that $c(X)$ (or $Nc(X)$) is not weakly contractible, or prove that no such $X$ exist.

nLab page on Nerve