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 $f \colon X \to Y$ 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 $\Gamma \subset Y$ of morphisms in $Y$ (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 $Y$ is uniquely a composition of those sequences of morphisms in $\Gamma$ 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 $f$ 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 $f \colon H \to K$ such that

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

  2. for each object $x$ of $H$ the induced morphism $H(x,x) \to K(f(x), f(x))$ 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 $f \colon H \to K$ such that

  1. for every object $x$ of $H$ and every morphism $\omega \colon f(x) \to y$ in $K_0$ there is a morphism $\hat \omega : x \to z$ of $H_0$ such that $f(\hat \omega) = \omega$;

  2. for every object $x$ in $H$ the induced morphism $f \colon H(x,x) \to K(f(x), f(x))$ 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):

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

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

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 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

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

hence we have a Quillen adjunction:

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

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

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]

A textbook account:

• 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]

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]