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:XY\mathbf{f} \colon \mathbf{X} \to \mathbf{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

    (1)ΓY \Gamma \;\subset\; \mathbf{Y}

    of morphisms in Y\mathbf{Y} (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 Y\mathbf{Y} is uniquely the composition of a reduced sequence of morphisms

      1. in Γ\Gamma,

      2. in the image under f\mathbf{f} of non-identity morphisms in X\mathbf{X},

      3. their inverses,

      where reduced means that:

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

      2. no two non-identity morphisms in the image of f\mathbf{f} 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):

(2)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 skeletal simplicial groupoids

The following proposition assumes the axiom of choice in the underlying category of Sets.


Every XsSet-Grpd\mathbf{X} \,\in\, sSet\text{-}Grpd has a bifibrant resolution by a skeletal groupoid.

Here we mean a skeletal sSet-enriched groupoid, hence a disjoint union of simplicial delooping groupoids (2).

By the existence of the model structure (Prop. ) all objects admit a cofibrant resolution. Moreover every sSetsSet-groupoid has a deformation retraction onto a skeleton (by this Prop., assuming the axiom of choice), and since cofibrancy is preserved by retraction it follows that every X\mathbf{X} has a cofibrant resolution by a skeleton. To conclude we just need to see that skeletal simplicial groupoids are fibrant, which is the case by Moore’s theorem.

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 simplicial sets. 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 retracts.

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 structures on simplicial groupoids and the Bergner-model structure on sSet-categories (see there),

hence we have a Quillen adjunction:

(3)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)

Consider the pair of adjoint functors given by homotopy coherent nerve NN and rigidification of quasi-categories \mathfrak{C}:

(4)sSet-Cat BergNsSet sSet\text{-}Cat_{Berg} \underoverset {\underset{N}{\longrightarrow}} {\overset{\mathfrak{C}}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet

This is a Quillen adjunction with respect to the Joyal model structure for quasi-categories on the right, but next we are after a slightly different role of this adjunction:


For SsSetS \in sSet there is a natural transformation

F(S)𝒢(S) F \circ \mathfrak{C}(S) \longrightarrow \mathcal{G}(S)

which is a Dwyer-Kan equivalence (from the localization (3) of the rigidification (4) to the Dwyer-Kan fundamental simplicial groupoid).

[Minichiello, Rivera & Zeinalian (2023), Thm. 1.1 (Cor. 4.2)]

As a corollary:


The composite of the adjunctions (3) and (4)

sSet-Grpd DKιFsSet-CatNsSet KQ sSet\text{-}Grpd_{DK} \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{F}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet\text{-}Cat \underoverset {\underset{N}{\longrightarrow}} {\overset{\mathfrak{C}}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet_{KQ}

is a Quillen adjunction between the Dwyer-Kan model structure from Prop. and the Kan-Quillen model structure on simplicial sets.


It is sufficient to see that FF \circ \mathfrak{C} is a left Quillen functor. This is immediate for the Joyal model structure on sSetsSet, where N\mathfrak{C} \dashv N is a Quillen adjunction (this Prop.) so that FF \circ \mathfrak{C}, being the composite of the left Quillen functor \mathfrak{C} with the left Quillen functor FF (3) is itself left Quillen. But the cofibrations of the Joyal model structure coincide with those of the Kan-Quillen model structure on sSet (both being monomorphisms), so that FF \circ \mathfrak{C} also preserves the Kan-Quillen cofibrations. To conclude it is therefore sufficient to see that FF \circ \mathfrak{C} sends all simplicial weak homotopy equivalences to Dwyer-Kan equivalences. But by Prop. every map f:SSf \colon S \longrightarrow S' of simplicial sets gives rise to a commuting diagram in sSet Cat of the form

F(S) F(f) F(S) 𝒢(S) 𝒢(f) 𝒢(S) \array{ F \circ \mathfrak{C}(S) &\overset{F \circ \mathfrak{C}(f)}{\longrightarrow}& F \circ \mathfrak{C}(S') \\ \Big\downarrow && \Big\downarrow \\ \mathcal{G}(S) &\underset{\;\; \mathcal{G}(f) \;\;}{\longrightarrow}& \mathcal{G}(S') }

where the vertical morphisms are Dwyer-Kan equivalences. Now observe that the Dwyer-Kan fundamental simplicial groupoid-functor 𝒢\mathcal{G} preserves all weak equivalences by Ken Brown's lemma: because it is a left Quillen functor on a model category all whose objects are cofibrant. Therefore also FF \circ \mathfrak{C} preserves all weak equivalences, by 2-out-of-3.


Fiber products with action 1-groupoids

While the underlying category of sSet-enriched groupoids is cartesian closed (see there), the model category is not cartesian monoidal as a model structure.

But here is a (very) simple special case at least of an internal hom Quillen adjunction for action 1-groupoids.


  • a group GGrp(Set)G \,\in\, Grp(Set),

  • with delooping groupoid denoted BGGrpd(Set)sSet-Grpd\mathbf{B}G \,\in\, Grpd(Set) \subset sSet\text{-}Grpd,

  • an inhabited set WSetW \,\in\, Set,

  • equipped with a group action GWGAct(Set)G \curvearrowright W \,\in\, G Act(Set),

  • with action groupoid denoted WGGrpdsSet-GrpdW \sslash G \,\in\, Grpd \hookrightarrow sSet\text{-}Grpd,

    canonically regarded in the slice sSet-Grpd /BGsSet\text{-}Grpd_{/\mathbf{B}G},

  • sSet-enriched groupoidsX,YsSet-Grpd\;\mathbf{X}, \mathbf{Y} \,\in\, sSet\text{-}Grpd lifted to the slice over BG\mathbf{B}G via p X,p YsSet-Grpd /BGp_\mathbf{X},\, p_\mathbf{Y} \,\in\, sSet\text{-}Grpd_{/\mathbf{B}G}.


With the above assumptions, for f:XY\mathbf{f} \colon \mathbf{X} \to \mathbf{Y} a morphism in the slice over BG\mathbf{B}G which is a cofibration in the slice model category in that its underlying morphism in sSet-GrpdsSet\text{-}Grpd is a Dwyer-Kan cofibration (Def. ), then also the fiber product morphism

(WG)× BGf:(WG)× BGX(WG)× BGY (W \sslash G) \times_{\mathbf{B}G} \mathbf{f} \;\colon\; (W \sslash G) \times_{\mathbf{B}G} \mathbf{X} \longrightarrow (W \sslash G) \times_{\mathbf{B}G} \mathbf{Y}

is a DK-(slice-)cofibration.


By definition, the cofibration f\mathbf{f} is a retraction (in the arrow category of sSet-GrpdsSet\text{-}Grpd) of a free map f^\widehat {f} (Def. ). But the right part of such a retraction diagram exhibits also f^\widehat{f} as morphism in the slice over BG\mathbf{B}G such that also the left part lifts to the slice: Regarded this way as a diagram in the slice, the functor (WG)× BG()(W \sslash G) \times_{\mathbf{B}G} (-) preserves its retraction property.

Therefore it is sufficient to prove the claim under the assumption that f\mathbf{f} is actually a free map.

So let ΓMor(Y)\Gamma \subset Mor(\mathbf{Y}) denote a set of generators which exhibits f\mathbf{f} as a free map. We claim that then the set

(WG)× BGΓMor((WG)× BGY) (W \sslash G) \times_{\mathbf{B}G} \Gamma \;\; \subset \;\; Mor\big( (W \sslash G) \times_{\mathbf{B}G} \mathbf{Y} \big)

serves as a set of generators exhibiting (WG)× BGf(W \sslash G) \times_{\mathbf{B}G} f as a free map, and hence as a cofibration.

Namely, by the nature of the fiber product, every morphism in (WG)× BGY(W \sslash G) \times_{\mathbf{B}G} \mathbf{Y} is of the form

(p Y(ϕ),ϕ):(w,y)(w,y), \big( p_{\mathbf{Y}}(\phi), \phi \big) \;\colon\; (w, y) \longrightarrow (w', y') \,,

where w=p Y(ϕ)ww' \,=\, p_{\mathbf{Y}}(\phi) \cdot w is determined by the pair consisting of ϕ:yy\phi \colon y \to y' and of wObj(Y)w \in Obj(\mathbf{Y}). By assumption on Γ\Gamma, ϕ\phi has a unique reduced decomposition into Γ\Gamma-components and this uniquely lifts to a reduced decomposition in (WG)× BGΓ(W \sslash G)\times_{\mathbf{B}G} \Gamma, determined by ww.


With the above assumptions, we have a Quillen adjunction on the slice model category of the DK-model structure:

sSet-Grpd /BG QuMap(WG,) BG(WG)× BG()sSet-Grpd /BG. sSet\text{-}Grpd_{/\mathbf{B}G} \underoverset { \underset{ Map\big( W \sslash G, - \big)_{\mathbf{B}G} }{\longleftarrow} } { \overset{ (W \sslash G) \times_{\mathbf{B}G} (-) }{\longrightarrow} } { \bot_{\mathrlap{Qu}} } sSet\text{-}Grpd_{/\mathbf{B}G} \,.


With Lem. it is now sufficient to show that (WG)× BG()(W \sslash G) \times_{\mathbf{B}G}(-) preserves all weak equivalences.

So assume that

f x,x:X(x,x)Y(f(x),f(x)) \mathbf{f}_{x,x'} \,\colon\, \mathbf{X}(x,x') \longrightarrow \mathbf{Y}\big(\mathbf{f}(x),\mathbf{f}(x')\big)

is a weak equivalence for all w,wObj(X)w,w' \,\in\, Obj(\mathbf{X}).

By the fact that f\mathbf{f} is a map in the slice over BG\mathbf{B}G, these component maps decompose into disjoint unions indexed by gGg \in G:

f x,x=gG(f x,x g:((p X) x,x) 1({g})((p Y) f(x),f(x)) 1({g})). \mathbf{f}_{x,x'} \;=\; \underset{g \in G}{\coprod} \Big( \mathbf{f}^g_{x,x'} \,\colon\, \big((p_{\mathbf{X}})_{x,x'}\big)^{-1} \big(\{g\}\big) \longrightarrow \big( (p_{\mathbf{Y}})_{ \mathbf{f}(x), \mathbf{f}(x') }\big)^{-1} \big(\{g\}\big) \Big) \,.

Accordingly, all the components f x,x g\mathbf{f}^g_{x,x'} must be weak equivalences. But then

((WG)× BGf) (w,x),(w,x)=gGgw=wf x,x g \big( (W \sslash G) \times_{\mathbf{B}G} \mathbf{f} \big)_{(w,x), (w',x') } \;=\; \underset{ {g \in G} \atop g \cdot w = w' }{\coprod} \mathbf{f}^g_{x,x'}

is a coproduct of weak equivalences and hence a weak equivalence.


The original article:

Textbook accounts:

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 October 31, 2023 at 14:54:31. See the history of this page for a list of all contributions to it.