nLab model structure on simplicial groupoids

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

Remark

(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).

Definition

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

Definition

(free morphisms of simplicial groupoids)
Say that a morphism $\mathbf{f} \colon \mathbf{X} \to \mathbf{Y}$ 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)$$\Gamma \;\subset\; \mathbf{Y}$$

   of morphisms in $\mathbf{Y}$ (of any degree) with the following properties:

   1. $\Gamma$ contains no identity morphisms;

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

   3. every non-identity morphism in $\mathbf{Y}$ is uniquely the composition of a reduced sequence of morphisms

      1. in $\Gamma$,

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

      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 $\mathbf{f}$ are consecutive.

[Dwyer & Kan 1984, §2.3]

Proposition

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

• weak equivalences are the Dwyer-Kan equivalences, hence those morphisms $\mathbf{f} \colon \mathbf{H} \to \mathbf{K}$ such that

  1. $\mathbf{f}$ induces in isomorphism on connected components $\pi_0 \mathbf{f} \colon \pi_0 \mathbf{H} \to \pi_0 \mathbf{K}$;

  2. for each object $x$ of $\mathbf{H}$ the induced morphism $\mathbf{H}(x,x) \to \mathbf{K}\big(\mathbf{f}(x), \mathbf{f}(x)\big)$ is a weak equivalence in the model structure on simplicial groups, which in turn equivalently means that it is a weak equivalence of underlying simplicial sets in the classical model structure on simplicial sets (a simplicial weak homotopy equivalence).

• fibrations are the Kan-isofibrations, namely those morphisms $\mathbf{f} \colon \mathbf{H} \to \mathbf{K}$ such that

  1. for every object $x$ of $\mathbf{H}$ and every morphism $\omega \colon \mathbf{f}(x) \to y$ in $\mathbf{K}_0$ there is a morphism $\hat \omega \colon x \to z$ of $H_0$ such that $\mathbf{f}(\hat \omega) = \omega$;

  2. for every object $x$ in $\mathbf{H}$ the induced morphism $\mathbf{f} \colon \mathbf{H}(x,x) \to \mathbf{K}\big(\mathbf{f}(x), \mathbf{f}(x)\big)$ is a Kan fibration.

• cofibrations are the retracts (in the arrow category) of the free maps from Def. .

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.

Remark

(relation to model structure on simplicial groups)
An $X \in sGrpd_{DK}$ for which $X_0 = \ast$ 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.

Properties

Extra model category properties

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

Definition

(simplicial interval groupoid)
Write

$$\mathcal{F} \;\colon\; sSet \overset{\mathcal{I}}{\longrightarrow} sSet\text{-}Cat \longrightarrow sSet\text{-}Grpd$$

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

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.)

Example

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

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

Proposition

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:

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

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

Proposition

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

This is Bergner (2008), Prop. 2.5.

Relation to simplicial groups

Remark

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

(2)$$\array{ sGrp &\hookrightarrow& sSet\text{-}Grpd \\ \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G} \mathrlap{\,.} }$$

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.

Proposition

Every $\mathbf{X} \,\in\, sSet\text{-}Grpd$ 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).

Proof

By the existence of the model structure (Prop. ) all objects admit a cofibrant resolution. Moreover every $sSet$-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 $\mathbf{X}$ 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

Proposition

The

• Dwyer-Kan loop groupoid-construction $\mathcal{G}$ (left adjoint)

• simplicial classifying space-construction $\overline{W}$ (right adjoint, see here)

constitute a Quillen equivalence

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

between the model structure on $sGrpd_{DK}$ from Prop. and the classical model structure on simplicial sets.

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

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

Remark

When restricted to simplicial groupoids of the form $(\mathbf{B} G)_\bullet$ for $G_\bullet$ a simplicial group and $\mathbf{B} G_n$ 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.

Lemma

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

One lazy way to see this:

Proof

For $f_\bullet \in Fib \cap \mathrm{W}$ 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 $\overline{W}(f) \in Fib \cap \mathrm{W}$ 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 $\overline{W}$ is the identity on the sets of objects/vertices (by its definition here), the claim follows.

Also useful to notice is:

Proposition

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

Proof

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

Proposition

The canonical inclusion functor

$$\iota \,\colon\, sSet\text{-}Grpd \xhookrightarrow{\phantom{--}} sSet\text{-}Cat$$

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

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

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

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

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:

Proposition

For $S \in sSet$ there is a natural transformation

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

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:

Proposition

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

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

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

Proof

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

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

where the vertical morphisms are Dwyer-Kan equivalences. Now observe that the Dwyer-Kan fundamental simplicial groupoid-functor $\mathcal{G}$ 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 $F \circ \mathfrak{C}$ preserves all weak equivalences, by 2-out-of-3.

Examples

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.

Consider:

• a group $G \,\in\, Grp(Set)$,

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

• an inhabited set $W \,\in\, Set$,

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

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

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

• sSet-enriched groupoids$\;\mathbf{X}, \mathbf{Y} \,\in\, sSet\text{-}Grpd$ lifted to the slice over $\mathbf{B}G$ via $p_\mathbf{X},\, p_\mathbf{Y} \,\in\, sSet\text{-}Grpd_{/\mathbf{B}G}$.

Lemma

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

$$(W \sslash G) \times_{\mathbf{B}G} \mathbf{f} \;\colon\; (W \sslash G) \times_{\mathbf{B}G} \mathbf{X} \longrightarrow (W \sslash G) \times_{\mathbf{B}G} \mathbf{Y}$$

is a DK-(slice-)cofibration.

Proof

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

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

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

$$(W \sslash G) \times_{\mathbf{B}G} \Gamma \;\; \subset \;\; Mor\big( (W \sslash G) \times_{\mathbf{B}G} \mathbf{Y} \big)$$

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

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

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

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

Proposition

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

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

Proof

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

So assume that

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

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

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

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

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

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

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

Related concepts

• Dwyer-Kan loop groupoid

• model structure on simplicial groups

• model structure on simplicial categories

• model structure on simplicial sets

• model structure on presheaves of simplicial groupoids

References

The original article:

• William Dwyer, Daniel Kan, Homotopy theory and simplicial groupoids, Indagationes Mathematicae (Proceedings) 87 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

Textbook accounts:

• Paul Goerss, J. F. Jardine, after corollary 7.3 in chapter V of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [doi:10.1007/978-3-0346-0189-4, webpage]

• John F. Jardine, §9.3 in: Local homotopy theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]

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

• Julia E. Bergner, Section 2 of: Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy Appl. 10 2 (2008) 175-193 [doi:10.4310/HHA.2008.v10.n2.a9, doi:0710.2254, erratum:doi:10.4310/HHA.2012.v14.n1.a15]

See also:

• A. R. Garzon, J. G. Miranda and R. Osorio, A simplicial description of the homotopy category of simplicial groupoids, Theory and Applications of Categories 7 14 (2000)263-283 [tac:7-14]

• Emilio Minichiello, Manuel Rivera, Mahmoud Zeinalian, Categorical models for path spaces, Advances in Mathematics 415 (2023) 108898 [arXiv:2201.03046, doi:10.1016/j.aim.2023.108898]