nLab geometric realization of categories

Idea

What is called geometric realization of categories is a functor that sends categories to topological spaces, namely the functor which first forms the simplicial set $N(\mathcal{C})$ that is the nerve of the category $\mathcal{C}$, and then forms the geometric realization ${\vert N(\mathcal{C})\vert}$ of this simplical set. Typically one is interested in this geometric realization up to weak homotopy equivalence.

By the homotopy hypothesis-theorem the geometric realization of simplicial sets constitutes a (Quillen)equivalence between the classical homotopy theory of simplicial sets and the classical homotopy theory of topological spaces. This means that inasmuch as one is interested in geometric realization of categories up to weak homotopy equivalence, then the key part of the operation is in forming the simplicial nerve $N(\mathcal{C})$ of a category, with the latter regarded as a model for an ∞-groupoid. Indeed, equivalently one could consider the Kan fibrant replacement of the nerve $N(\mathcal{C})$ (which still has the same geometric realization, up to weak homotopy equivalence).

Therefore an equivalent perspective on geometric realization of categories is that it universally turns a category into an infinity-groupoid by freely turning all its morphisms into homotopy equivalences.

Geometric realization of categories has various good properties:

It sends equivalences of categories to weak homotopy equivalences (corollary below). A more general sufficient criterion for the geometric realization of a functor is given by the seminal theorem known as Quillen’s theorem A (theorem below.)

The existence of the Thomason model structure (below) implies that every homotopy type arises as the geometric realization of some category. In fact it already arises as the geometric realization of some poset ((0,1)-category).

Definition

Write

$$N \colon Cat \to sSet$$

for the nerve functor from Cat to sSet. Write

$${\vert - \vert} : sSet \to Top$$

for the geometric realization of simplicial sets from sSet to Top.

The geometric realization of categories is the composite of these two operations:

$${\vert - \vert} \coloneqq {\vert N(-)\vert} \;\colon\; Cat \to Top$$

Properties

Thomason model structure

There is a model category structure on Cat whose weak equivalences are those functors $F \colon \mathcal{C} \to \mathcal{D}$ which under geometric realization become weak equivalences in the classical model structure on topological spaces, hence which become weak homotopy equivalences. This is called the Thomason model structure.

The existence of the Thomas model structure implies that every homotopy type arises as the geometric realization of some category, in fact already as the realization of some poset/(0,1)-category:

Definition

For $C$ a category, let $\nabla C$ be the poset of non-degenerate simplices in the subdivision of the nerve $N C$, ordered by inclusion.

Proposition

For every category $\mathcal{C}$ the poset $\nabla \mathcal{C}$ from def. has weakly homotopy equivalent geometric realization

$${\vert N(\nabla \mathcal{C}) \vert} \;\simeq_{wh}\; {\vert \mathcal{C} \vert} \,.$$

(e.g. by this Prop.)

Recognizing weak equivalences: Quillen’s theorem A and B

Let $\mathcal{C}, \mathcal{D}$ be two categories and let

$$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$$

be a functor between them.

Theorem

(Quillen 72, theorem A)

If for all objects $d \in \mathcal{D}$ the geometric realization ${\vert N(F/d)\vert}$ of the comma category $F/d$ is contractible (meaning that $F$ is a “homotopy cofinal functor”, hence a cofinal (∞,1)-functor), then

$${\vert N(F) \vert} \;\colon\; {\vert N(\mathcal{C}) \vert} \longrightarrow {\vert N(\mathcal{D}) \vert}$$

is a weak homotopy equivalence.

Theorem

(Quillen 72 theorem B)

If for all objects $d \in \mathcal{D}$ we have that ${\vert N(F/d)\vert}$ is weakly homotopy equivalent to a given topological space $X$ and all morphisms $f \colon d_1 \to d_2$ induce weak homotopy equivalences between these, then $X$ is the homotopy fiber of ${\vert N(F) \vert}$, hence we have a homotopy fiber sequence (in the classical model structure on topological spaces) of the form

$$X \longrightarrow {\vert N(\mathcal{C}) \vert} \overset{\vert N(F) \vert }{\longrightarrow} {\vert N(\mathcal{D}) \vert} \,.$$

As a consequence:

Proposition

(McCord 66, theorem 6, Quillen 78, prop. 1.6)

Let $\mathcal{C}, \mathcal{D}$ be finite posets and consider $F \colon \mathcal{C} \to \mathcal{D}$ be a functor.

If for each element/object $y \in \mathcal{D}$ its preimage $f^{-1}( \{ y' \in Y \vert y' \leq y \})$ has contractible geometric realization, then ${\vert N(F)\vert}$ is a homotopy equivalence.

An alternative proof is given in (Barmak 10).

Quillen’s Theorem B for Grothendieck fibrations

Quillen’s theorem B raises the following question: Which pullback square

of small categories does the functor $|N(-)|$ carry to a homotopy pullback square? One such instance is where $p$ is a Grothendieck fibration which induces homotopy equivalences between the classifying spaces. This appears as (LM15, Theorem 2.7). A version for (infinity,1)-categories appears in (Arakawa24, Proposition 2.26) and (KSW24, Lemma A.1). More discussions can be found in (Cisinski19, Section 4.6).

Natural transformations and homotopies

Proposition

A natural transformation $\eta : F \Rightarrow G$ between two functors $F, G : \mathcal{C} \to \mathcal{D}$ induces under geometric realization a homotopy

$${|N(\eta)|} \colon {\vert N(F)\vert} \longrightarrow {\vert N(G) \vert} \,.$$

Proof

The natural transformation is equivalently a functor of the form

$$\eta \;\colon\; \mathcal{C} \times \{0 \to 1\} \to \mathcal{D}$$

out of the product category of $\mathcal{C}$ with the interval category.

Since geometric realization of simplicial sets preserves Cartesian products (see there) we have that

$${\vert N( \mathcal{C} \times \{0,1\} ) \vert} \;\simeq_{iso}\; {\vert N(\mathcal{C}) \vert} \times {\vert N(\{0 \to 1\}) \vert}$$

But this is a cylinder object in topological spaces, hence ${\vert N(\eta) \vert}$ is a left homotopy.

Corollary

An equivalence of categories $\mathcal{C} \simeq \mathcal{D}$ induces a homotopy equivalence between their geometric realizations.

Remark

The statement still remains true for a pair of adjoint functors $\mathcal{C} \leftrightarrows \mathcal{D}$.

Remark

Notice that the converse is far from true: Very different categories can have geometric realizations that are (weakly) homotopy equivalent. This is because geometric realization implicitly involves Kan fibrant replacement: it freely turns morphisms into equivalences.

Corollary

If a category $\mathcal{C}$ has an initial object or a terminal object, then its geometric realization is contractible.

Proof

Assume the case of a terminal object, the other case works formally dually. Write $*$ for the terminal category.

Then we have an equality of functors

$$Id_* = (* \stackrel{\bottom}{\to} C \to *) \,,$$

where the first functor on the right picks the terminal object, and we have a natural transformation

$$Id_C \Rightarrow (C \to * \stackrel{\bottom}{\to} C)$$

whose components are the unique morphisms into the terminal object.

By prop. it follows that we have a homotopy equivalence $\vert N(\mathcal{C}) \vert \to \vert N(\ast) \vert = \ast$.

Behaviour under homotopy colimits

Proposition

For $F \colon \mathcal{D} \to Cat$ a functor, let

$${\vert N(F(-))\vert} \;\colon\; \mathcal{D} \overset{F}{\longrightarrow} Cat \stackrel{\vert N(-) \vert}{\to} Top$$

be its postcomposition with geometric realization of categories

Then we have a weak homotopy equivalence

$${\left\vert N\left(\int F \right) \right\vert} \simeq hocolim {\vert N(F(-)) \vert}$$

exhibiting the homotopy colimit in Top over $\vert N(F (-)) \vert$ as the geometric realization of the Grothendieck construction $\int F$ of $F$.

This is due to (Thomason 79).

Related concepts

• geometric realization

  • of categories, of simplicial topological spaces, of cohesive ∞-groupoids

References

General

For general references see also nerve and geometric realization.

Quillen’s theorems A and B

The original articles are

• Michael C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), 465-474 (pdf)

• Daniel Quillen, Higher algebraic K-theory, I: Higher K-theories Lect. Notes in Math. 341 (1972), 85-1 (pdf)

• Daniel Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), 101-128.

The geometric realization of Grothendieck constructions has been analyzed in

• R. W. Thomason, Homotopy colimits in the category of small categories , Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91109.(pdf)

Review is in

• Jonathan Barmak, On Quillen’s Theorem A for posets, Journal of Combinatorial Theory Series A, Volume 118 Issue 8, November, 2011

  Pages 2445-2453 (arXiv:1005.0538)

Further development includes

• Clark Barwick, Daniel Kan, A Quillen theorem $B_n$ for homotopy pullbacks (arXiv:1101.4879)

• David Roberts, Homotopy types of topological stacks of categories, New York Journal of Mathematics, Volume 30 (2024), 940-955, journal version, arXiv:2204.02778

For variations of Quillens’ Theorem B and its generalizations for (infinity,1)-categories:

• Zhen Lin Low, Aaron Mazel-Gee, From fractions to complete Segal spaces, Homology Homotopy Appl. 17 1 (2015) 321–338 [arXiv:1409.8192, doi:10.4310/HHA.2015.v17.n1.a16]

• Denis-Charles Cisinski, Section 4.6 of Higher categories and homotopical algebra, Cambridge Studies in Advanced Mathematics, 2021, [available at the author’s webpage, doi:10.1017/9781108588737]

• Kensuke Arakawa, A Context for Manifold Calculus, arXiv:2403.03321

• Eilind Karlsson?, Claudia Scheimbauer, Tashi Walde?, Assembly of Constructible Factorization Algeras, arXiv:2403.19472