nLab model structure on simplicial groupoids



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Enriched category theory


Before we start, beware the usual terminology issue with “simplicial groupoids”:


(terminology) This entry is concernd with simplicial groupoids as traditionally understood (following Dwyer & Kan (1984)), referring to simplicial objects in the category Grpd of groupoids with the special property that their simplicial set of objects is simplicially constant. Any such “Dwyer-Kan simplicial groupoid” is equivalently an sSet-enriched category that is a groupoid in the enriched sense: an sSet-enriched groupoid. Therefore, and since in applications it is often this sSet-enriched structure which matters, a more accurate term would be simplicially enriched groupoids, but this terminology is not at all standard. See the corresponding discussion at simplicial groupoid (here).


Write sGrpd DKsGrpd_{DK} for the category of simplicial groupoids (whose simplicial sets of objects are understood to be constant, see the discussion there).


(free morphisms of simplicial groupoids)
Say that a morphism f:XYf \colon X \to Y of simplicial groupoids is free iff:

  1. it is degreewise injective (i.e. on the sets of objects and on the sets of morphisms in each degree);

  2. there is a subset ΓY\Gamma \subset Y of morphisms in YY (of any degree) with the following properties:

    1. Γ\Gamma contains no identity morphisms;

    2. Γ\Gamma is closed under forming degenerate cells;

    3. every non-identity morphism in YY is uniquely a composition of those sequences of morphisms in Γ\Gamma and their inverses which are reduced in that:

      1. no morphism in the sequences is consecutive with its inverse,

      2. no two non-identity morphisms in the image of ff are consecutive.

[Dwyer & Kan 1984, §2.3]


(Dwyer-Kan model structure on simplicial groupoids)
There is a model category structure on sGrpd DKsGrpd_{DK} whose

This is due to Dwyer & Kan (1984), §1.1, §1.2, Thm. 2.5, reviewed in Goerss & Jardine (2009), p. 316 and after cor V.7.3.


(relation to model structure on simplicial groups)
An XsGrpd DKX \in sGrpd_{DK} for which X 0=*X_0 = \ast is the terminal groupoid is, when regarded as a pointed object, equivalently the (delooping of the) simplicial group which is its unique hom-object. Restricted to such “simplicial delooping groupoids” of simplicial groups and under this identitification, the fibrations and weak equivalences in Prop. are those of the model structure on simplicial groups.


Extra model category properties

We write sSet-GrpdsSet\text{-}Grpd for the category of Dwyer-Kan simplicial groupoids.


(simplicial interval groupoid)

:sSetsSet-CatsSet-Grpd \mathcal{F} \;\colon\; sSet \overset{\mathcal{I}}{\longrightarrow} sSet\text{-}Cat \longrightarrow sSet\text{-}Grpd

for the functor which sends a simplicial set SS to the sSet-enriched groupoid (S)\mathcal{F}(S) which has precisely two objects 0,10,1, no non-trivial endomorphisms and isomorphisms between 00 and 11 freely generated from the cells of XX.

This is Dwyer & Kan (1984), §2.8, related to the Milnor construction in Goerss & Jardine (2009), pp. 314. (Beware that in Bergner (2008), p. 4 the statement of free generation is missing.)


The construction of Def. applied to the terminal simplicial set Δ[0]\Delta[0] (0-simplex, “singleton”) is the interval groupoid:

(*)={01}. \mathcal{F}(\ast) \,=\, \big\{ 0 \overset{\sim}{\leftrightarrows} 1 \big\} \,.


The model structure on simplicial groupoids from Prop. is cofibrantly generated with generating (acyclic) cofibrations the images under the simplicial interval functor (Def. ) of the generating (acyclic) cofibrations in the classical model structure on simplicial sets (see there), hence of the boundary- and horn-inclusions of simplices, respectively:

I{(i n):(Δ[n])(Δ[n])} n J{(j n k):(Λ k[n])(Λ k[n])} n +,0kn \begin{array}{l} I \,\coloneqq\, \Big\{ \mathcal{F}(i_n) \,\colon\, \mathcal{F}\big(\partial \Delta[n]\big) \hookrightarrow \mathcal{F}\big(\Delta[n]\big) \Big\}_{n \in \mathbb{N}} \\ J \,\coloneqq\, \Big\{ \mathcal{F}(j^k_n) \,\colon\, \mathcal{F}\big(\Lambda_k[n]\big) \hookrightarrow \mathcal{F}\big(\Lambda_k[n]\big) \Big\}_{n \in \mathbb{N}_+, 0 \leq k \leq n} \end{array}

This is essentially Dwyer & Kan (1984), Prop. 2.9, 2.10, made more explicit in Bergner (2008), Thm. 2.2.


The model structure on simplicial groupoids from Prop. is right proper.

This is Bergner (2008), Prop. 2.5.

Relation to simplicial groups


Forming simplicial delooping groupoids constitutes a fully faithful functor of 1-categories from simplicial groups to sSet-enriched groupoids (DK-simplicial groupoids):

sGrp sSet-Grpd 𝒢 B𝒢. \array{ sGrp &\hookrightarrow& sSet\text{-}Grpd \\ \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G} \mathrlap{\,.} }

With respect to the above model structure (Prop. ) this functor clearly preserves the weak equivalences and fibrations of the model structure on simplicial groups. However, it does not have a left adjoint and thus fails to be a right Quillen functor.

Relation to simplicial sets



constitute a Quillen equivalence

sGrpd DK QuW¯𝒢sSet Qu sGrpd_{DK} \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{\mathcal{G}}{\longleftarrow}} {\;\;\;\; \simeq_{_{\mathrlap{Qu}}} \;\;\;\;} sSet_{Qu}

between the model structure on sGrpd DKsGrpd_{DK} from Prop. and the classical model structure on simplicial sets.

In addition both 𝒢\mathcal{G} and W¯\overline W preserve all weak equivalences.

This is due to Dwyer & Kan (1984), Thm. 3.3, reviewed in Goerss & Jardine (2009), Thm. 7.8.


When restricted to simplicial groupoids of the form (BG) (\mathbf{B} G)_\bullet for G G_\bullet a simplicial group and BG n\mathbf{B} G_n its delooping groupoid this produces a standard presentation of looping and delooping for infinity-groups. See there and at model structure on simplicial groups for more.


Any acyclic fibration fFibWf \in Fib \cap \mathrm{W} of simplicial groupoids is surjective on objects.

One lazy way to see this:

For f FibWf_\bullet \in Fib \cap \mathrm{W} an acyclic Kan fibration, notice that:

  1. Since the simplicial classifying space functor on simplicial groupoids (here) is a right Quillen functor by Prop. it follows that W¯(f)FibW\overline{W}(f) \in Fib \cap \mathrm{W} is an acyclic Kan fibration

  2. All acylic fibrations in the classical model structure on simplicial sets are degreewise surjective (see this Prop.).

But since W¯\overline{W} is the identity on the sets of objects/vertices (by its definition here), the claim follows.

Also useful to notice is:


A (acyclic) cofibration of simplicial groupoids is hom-object-wise an (acyclic) injection, hence a Kan-Quillen (acyclic) cofibration, of simplicial hom-sets.


By Def. and Prop. these cofibrations of simplicial groupoids are, in particular, retracts of monomorphisms of simplcial set. Since sSet is a topos, its epi-mono factorization system implies that monomorphisms are preserved under retracts. And of course also the Kan-Quillen weak equivalences are preserved under retract.

Relation to simplicial categories


The canonical inclusion functor

ι:sSet-GrpdsSet-Cat \iota \,\colon\, sSet\text{-}Grpd \xhookrightarrow{\phantom{--}} sSet\text{-}Cat

of the category of sSet-enriched groupoids into that of sSet-enriched categories

  1. has a left adjoint, given degreewise by the free groupoid-construction (localization at the class of all morphisms)

  2. evidently preserves fibrations and weak equivalences between the above model structure on simplicial groupoids and the Bergner-model structure on sSet-categories (see there)

hence we have a Quillen adjunction:

sSet-Grpd DK QuιFsSet-Cat Berg. sSet\text{-}Grpd_{DK} \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{F}{\longleftarrow}} {\;\;\; \bot_{\mathrm{Qu}} \;\;\;} sSet\text{-}Cat_{Berg} \,.

(see also Minichiello, Rivera & Zeinalian (2023), Prop. 2.8)


The original article:

A textbook account:

A proof of the model structure closer to that establishing the model structure on simplicial categories and making explicit the cofibrant generation:

See also:

Last revised on May 14, 2023 at 08:19:56. See the history of this page for a list of all contributions to it.