model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
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 $sGrpd_{DK}$ 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 $\mathbf{f} \colon \mathbf{X} \to \mathbf{Y}$ of simplicial groupoids is free iff:
it is degreewise injective (i.e. on the sets of objects and on the sets of morphisms in each degree);
there is a subset
of morphisms in $\mathbf{Y}$ (of any degree) with the following properties:
$\Gamma$ contains no identity morphisms;
$\Gamma$ is closed under forming degenerate cells;
every non-identity morphism in $\mathbf{Y}$ is uniquely the composition of a reduced sequence of morphisms
in $\Gamma$,
in the image under $\mathbf{f}$ of non-identity morphisms in $\mathbf{X}$,
their inverses,
where reduced means that:
no morphism in the sequences is consecutive with its inverse,
no two non-identity morphisms in the image of $\mathbf{f}$ are consecutive.
(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
$\mathbf{f}$ induces in isomorphism on connected components $\pi_0 \mathbf{f} \colon \pi_0 \mathbf{H} \to \pi_0 \mathbf{K}$;
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
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 $\mathbf{f}(\hat \omega) = \omega$;
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. .
(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.
We write $sSet\text{-}Grpd$ for the category of Dwyer-Kan simplicial groupoids.
(simplicial interval groupoid)
Write
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 $X$.
The construction of Def. applied to the terminal simplicial set $\Delta[0]$ (0-simplex, “singleton”) is the interval groupoid:
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:
The model structure on simplicial groupoids from Prop. is right proper.
Forming simplicial delooping groupoids constitutes a fully faithful functor of 1-categories from simplicial groups to sSet-enriched groupoids (DK-simplicial groupoids):
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.
The following proposition assumes the axiom of choice in the underlying category of Sets.
Every $\mathbf{X} \,\in\, sSet\text{-}Grpd$ has a bifibrant resolution by a skeletal groupoid.
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.
The
Dwyer-Kan loop groupoid-construction $\mathcal{G}$ (left adjoint)
simplicial classifying space-construction $\overline{W}$ (right adjoint, see here)
constitute a Quillen equivalence
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.
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.
Any acyclic fibration $f \in Fib \cap \mathrm{W}$ of simplicial groupoids is surjective on objects.
For $f_\bullet \in Fib \cap \mathrm{W}$ an acyclic Kan fibration, notice that:
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
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:
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.
The canonical inclusion functor
of the category of sSet-enriched groupoids into that of sSet-enriched categories
has a left adjoint, given degreewise by the free groupoid-construction (localization at the class of all morphisms)
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:
Consider the pair of adjoint functors given by homotopy coherent nerve $N$ and rigidification of quasi-categories $\mathfrak{C}$:
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 $S \in sSet$ there is a natural transformation
which is a Dwyer-Kan equivalence (from the localization (3) of the rigidification (4) to the Dwyer-Kan fundamental simplicial groupoid).
As a corollary:
The composite of the adjunctions (3) and (4)
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 $F \circ \mathfrak{C}$ 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
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.
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 $p_\mathbf{X},\, p_\mathbf{Y} \,\in\, sSet\text{-}Grpd_{/\mathbf{B}G}$.
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
is a DK-(slice-)cofibration.
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
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
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 $w$.
With the above assumptions, we have a Quillen adjunction on the slice model category of the DK-model structure:
With Lem. it is now sufficient to show that $(W \sslash G) \times_{\mathbf{B}G}(-)$ preserves all weak equivalences.
So assume that
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 $g \in G$:
Accordingly, all the components $\mathbf{f}^g_{x,x'}$ must be weak equivalences. But then
is a coproduct of weak equivalences and hence a weak equivalence.
The original article:
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:
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]
Last revised on October 31, 2023 at 14:54:31. See the history of this page for a list of all contributions to it.