nLab geometric realization of categories



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Category theory



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(𝒞)N(\mathcal{C}) that is the nerve of the category 𝒞\mathcal{C}, and then forms the geometric realization |N(𝒞)|{\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(𝒞)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(𝒞)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).



N:CatsSet N \colon Cat \to sSet

for the nerve functor from Cat to sSet. Write

||:sSetTop {\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:

|||N()|:CatTop {\vert - \vert} \coloneqq {\vert N(-)\vert} \;\colon\; Cat \to Top


Thomason model structure

There is a model category structure on Cat whose weak equivalences are those functors F:𝒞𝒟F \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:


For CC a category, let C\nabla C be the poset of simplices in the nerve NCN C, ordered by inclusion.


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

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

Recognizing weak equivalences: Quillen’s theorem A and B

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

F:𝒞𝒟 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

be a functor between them.


(Quillen 72, theorem A)

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

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

is a weak homotopy equivalence.


(Quillen 72 theorem B)

If for all objects d𝒟d \in \mathcal{D} we have that |N(F/d)|{\vert N(F/d)\vert} is weakly homotopy equivalent to a given topological space XX and all morphisms f:d 1d 2f \colon d_1 \to d_2 induce weak homotopy equivalences between these, then XX is the homotopy fiber of |N(F)|{\vert N(F) \vert}, hence we have a homotopy fiber sequence (in the classical model structure on topological spaces) of the form

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

As a consequence:


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

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

If for each element/object y𝒟y \in \mathcal{D} its preimage f 1({yY|yy})f^{-1}( \{ y' \in Y \vert y' \leq y \}) has contractible geometric realization, then |N(F)|{\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()||N(-)| carry to a homotopy pullback square? One such instance is where pp 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


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

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

The natural transformation is equivalently a functor of the form

η:𝒞×{01}𝒟 \eta \;\colon\; \mathcal{C} \times \{0 \to 1\} \to \mathcal{D}

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

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

|N(𝒞×{0,1})| iso|N(𝒞)|×|N({01})| {\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 |N(η)|{\vert N(\eta) \vert} is a left homotopy.


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


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


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.


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


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 *=(*C*), 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(C*C) 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 |N(𝒞)||N(*)|=*\vert N(\mathcal{C}) \vert \to \vert N(\ast) \vert = \ast.

Behaviour under homotopy colimits


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

|N(F())|:𝒟FCat|N()|Top {\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

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

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

This is due to (Thomason 79).



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

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

Last revised on June 18, 2024 at 08:31:42. See the history of this page for a list of all contributions to it.