Under rather general conditions a functor
into a cocomplete category (possibly a -enriched category with some complete symmetric monoidal category) induces a pair of adjoint functors
between and the category of presheaves on (here = Set for the unenriched case)
We place ourselves in the context of -enriched category theory. The reader wishing to stick to the ordinary notions in locally small categories takes = Set.
The realization operation is the left Kan extension of along the Yoneda embedding (i.e. the Yoneda extension):
If we assume that is tensored over , then by the general coend formula for left Kan extension we find that for we have
For instance when is the simplex category this reads more recognizably
The corresponding nerve operation
is given by
Using the fact that the Hom in its first argument sends coends to ends and then using the definition of tensoring over , we check the hom-isomorphism
where in the last step we used the definition of the enriched functor category in terms of an end.
Topological realization of simplicial sets
A classical example is given by the cosimplicial topological space
that sends the abstract -simplex to the standard topological -simplex .
This topological nerve and realization adjunction plays a central role as a presentation of the Quillen equivalence between the model structure on simplicial sets and the model structure on topological spaces. This is discussed in detail at homotopy hypothesis.
Nerve and realization of categories
Pretty much every notion of category and higher category comes, or should come, with its canonical notion of simplicial nerve, induced from a functor
that sends the standard -simplex to something like the free -category on the -directed graph underlying that simplex.
For ordinary categories see the discussion at nerve and at geometric realization of categories.
One formalization of this for in the context of strict ω-categories is the cosimplicial -category called the orientals
The Dold–Kan correspondence is the nerve/realization adjunction for the homology functor
to the category of chain complexes of abelian groups, which sends the standard -simplex to its homology chain complex, more precisely to its normalized Moore complex.
Simplicial models for -categories
The canonical cosimplicial simplicially enriched category
induces the homotopy coherent nerve of SSet-enriched categories and establishes the relation between the quasi-category and the simplicially enriched model for (infinity,1)-categories. See
Full and faithfulness
Under some conditions one can characterize when and where the nerve construction is a full and faithful functor. For the moment see for instance monad with arities.
The notion of nerve and realization (not with these names yet) was introduced and proven to be an adjunction in section 3 of
- Daniel Kan, Functors involving c.s.s complexes, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330–346 (jstor).
In fact, in that very article apparently what is now called Kan extension is first discussed.
Also, in that article, as an example of the general mechanism, also the Dold–Kan correspondence was found and discussed, independently of the work by Dold and Puppe shortly before, who used a much less general-nonsense approach.
In an article in 1984, Dwyer and Kan look at this ‘nerve-realization’ context from a different viewpoint, using the term ‘singular functor’ where the above has used ‘nerve’. Their motivation example is that in which is the orbit category of a group , and the realisation starts with a functor on that category with values in spaces and returns a -space:
- W. G. Dwyer and D. M. Kan, Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 – 153.
We should also mention the treatment in Leinster’s book and the relation to the notions of dense subcategory or adequate subcategory in the sense of Isbell.