Contents

# Contents

## Idea

Under rather general conditions a functor

$S_C \;\colon\; S \to C$

into a cocomplete category $C$ (possibly a $V$-enriched category with $V$ some complete symmetric monoidal category) induces a pair of adjoint functors

$C \stackrel{\xleftarrow{|-|}}{\underset{N}{\rightarrow}} [S^{op}, V],$

where $|-| \dashv N$, between $C$ and the category of presheaves $PSh(S) = [S^{op}, V]$ on $S$ (here $V$ = Set for the unenriched case)

where

• $N$ behaves like a nerve operation;

• $|-|$ behaves like geometric realization.

###### Remark

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.

## Definition

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):

$\array{ S &\stackrel{S_C}{\to}&& C \\ \downarrow^{Y} &\Downarrow& \nearrow_{|-|} \\ [S^{op},V] } \,.$

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

$|X| \simeq \int^{s \in S} S_C(s) \cdot X_s \,.$

For instance when $S = \Delta$ is the simplex category this reads more recognizably

$|X| \simeq \int^{[n] \in \Delta} \Delta_C[n] \cdot X_n \,.$

The corresponding nerve operation

$N : C \stackrel{}{\to} [S^{op},V]$

is given by the restricted Yoneda embedding

$N(c) : S^{op} \stackrel{S_C^{op}}{\to} C^{op} \stackrel{C(-,c)}{\to} V \,.$

###### Theorem

Nerve and realization are a pair of adjoint functors

${|-|} \,\dashv\, N$

with $N$ the right adjoint.

(Kan 1958, Sec. 3)

###### Proof

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:

\begin{aligned} Hom_C(|X|, c) & \coloneqq Hom_C( \int^{s} S_C(s) \cdot X_s, c) \\ & \simeq \int_{s} Hom_C( S_C(s) \cdot X_s, c) \\ & \simeq \int_{s} Hom_V( X_s , C(S_C(s), c)) \\ & = \int_{s} Hom_V( X_s , N(c)_s) \\ & \simeq Hom_{[S^{op},V]}(X, N(c)) \,, \end{aligned}

where in the last step we used the definition of the enriched functor category in terms of an end.

###### Remark

In many cases we have $V =$ Set and the tensoring of an object $c$ over a set $I$ is given by coproducts as

$c \cdot I = \coprod_{i \in I} c \,.$

This is the case for instance for the below examples of realization of simplicial sets, nerves of categories and the Dold-Kan correspondence.

## Examples

### Topological realization of simplicial sets

A classical example is given by the cosimplicial topological space

$\Delta_{Top} : \Delta \to Top$

that sends the abstract $n$-simplex $[n]$ to the standard topological $n$-simplex $\Delta_{Top}[n] \subset \mathbb{R}^n$.

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

$\Delta_C : \Delta \to n Cat$

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 ∞-categories is the cosimplicial $\omega$-category called the orientals

$\Delta_{\omega} : \Delta \to \omega Cat \,.$
• The induced nerve is the ∞-nerve.

• The induced realization operation is the operation of forming the free $\omega$-category on a simplicial set. See oriental for more details.

### Dold–Kan correspondence

The Dold-Kan correspondence is the nerve/realization adjunction for the homology functor

$\Delta_{C_\bullet} : \Delta \to Ch_+$

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 induced realization is the normalized Moore complex functor extended from $\Delta$ to all simplicial sets.

### Simplicial models for $(\infty,1)$-categories

The canonical cosimplicial simplicially enriched category

$\Delta \to SSet\text{-}Cat$

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

## Properties

### 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.

## References

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:1993103).

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:

• 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.

In a blog post on the n-Category Café, Tom Leinster illustrates that “sections of a bundle” is a nerve operation, and its corresponding geometric realization is the construction of the étalé space of a presheaf.

Last revised on August 25, 2021 at 05:40:10. See the history of this page for a list of all contributions to it.