CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Under rather general conditions a functor
into a cocomplete category $C$ (possibly a $V$-enriched category with $V$ some complete symmetric monoidal category) induces a pair of adjoint functors
between $C$ and the category of presheaves $PSh(C) = [S^{op}, V]$ on $S$ (here $V$ = Set for the unenriched case)
where
$N$ behaves like a nerve operation;
$|-|$ behaves like geometric realization.
Here “$S$” is supposed to be suggestive of a category of certain “geometric Shapes”. The canonical example is $S = \Delta$, the simplex category, and the reader may find it helpful to keep that example in mind.
We place ourselves in the context of $V$-enriched category theory. The reader wishing to stick to the ordinary notions in locally small categories takes $V$= Set.
The realization operation is the left Kan extension of $S_C : S \to C$ along the Yoneda embedding $S \hookrightarrow [S^{op},V]$ (i.e. the Yoneda extension):
If we assume that $C$ is tensored over $V$, then by the general coend formula for left Kan extension we find that for $X \in [S^{op}, V]$ we have
For instance when $S = \Delta$ is the simplex category this reads more recognizably
The corresponding nerve operation
is given by
Nerve and realization are a pair of adjoint functors
with $N$ right adjoint.
Using the fact that the Hom in its first argument sends coends to ends and then using the definition of tensoring over $V$, we check the hom-isomorphism
where in the last step we used the definition of the enriched functor category in terms of an end.
In many cases we have $V =$ Set and the tensoring of an object $c$ over a set $I$ is given by coproducts as
This is the case for instance for the below examples of realization of simplicial sets, nerves of categories and the Dold-Kan correspondence.
A classical example is given by the cosimplicial topological space
that sends the abstract $n$-simplex $[n]$ to the standard topological $n$-simplex $\Delta_{Top}[n] \subset \mathbb{R}^n$.
The corresponding realization is what is traditionally called geometric realization of simplicial sets.
By restricting this to simplicial sets which are themselves simplicial nerves of categories (see below) or more generally are quasi-categories, this also induces the notion of geometric realization of categorical structures.
The construction generalizes also to a notion of geometric realization of simplicial topological spaces.
The corresponding nerve is the singular simplicial complex functor, producing the fundamental ∞-groupoid of a topological space.
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.
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 $n$-simplex to something like the free $n$-category on the $n$-directed graph underlying that simplex.
For ordinary categories see the discussion at nerve and at geometric realization of categories.
One formalization of this for $n = \infty$ in the context of strict ω-categories is the cosimplicial $\omega$-category called the orientals
The induced realization operation is the operation of forming the free $\omega$-category on a simplicial set. See oriental for more details.
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 $n$-simplex to its homology chain complex, more precisely to its normalized Moore complex.
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
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
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 $S$ is the orbit category of a group $G$, and the realisation starts with a functor on that category with values in spaces and returns a $G$-space:
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.