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
Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
Depending on the chosen model category structure, the category sSet of simplicial sets may model ∞-groupoids (for the standard model structure on simplicial sets) or (∞,1)-categories in the form of quasi-categories (for the Joyal model structure) on SSet.
Accordingly, there are model category structures on sSet-categories that similarly model (n,r)-categories with $r$ shifted up by 1:
This we discuss below.
there is another model structure on sSet-categories whose fibrant objects are (∞,1)-category/quasi-category-enriched categories, and which model (∞,2)-categories.
For more on this see elsewhere
Both are special cases of a model structure on enriched categories.
Here we describe the model category structure on SSet Cat that makes it a model for the (∞,1)-category of (∞,1)-categories.
(Dwyer-Kan equivalences)
An sSet-enriched functor $F : C \to D$ between sSet-categories is called a weak equivalence precisely if
it is essentially surjective in that the induced functor of homotopy categories is an ordinary essentially surjective functor;
it is an $\infty$-full and faithful functor in that for all objects $x,y \in C$ the morphism
is a weak equivalence in the standard model structure on simplicial sets.
This notion is due to Dwyer & Kan (1980 FuncComp), §2.4 and now known as Dwyer-Kan equivalences.
A Quillen equivalence $D \leftrightarrows C$ between model categories induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ between their simplicial localizations.
This is made explicit in Mazel-Gee 15, p. 17 to follow from Dwyer & Kan 80, Prop. 4.4 with 5.4.
The analogous statement under further passage to Joyal equivalences of quasi-categories is Lurie (2009), Cor. A.3.1.12, under the additional assumptions that the model categories are simplicial, that every object of $C$ is cofibrant and that the right adjoint is an sSet enriched functor.
The category SSet Cat of small sSet-enriched categories carries the structure of a model category with
weak equivalences the Dwyer-Kan equivalences (Def. );
fibrations those sSet-enriched functors $F \colon C \to D$ such that
for all $x, y \in C$ the morphism $F_{x,y} \colon C(x,y) \to D\big(F(x), F(y)\big)$ is a fibration in the classical model structure on simplicial sets;
the induced functor $\pi_0(F) \colon Ho(C) \to Ho(D)$ on homotopy categories is an isofibration.
[Bergner (2004), Lurie (2009), thm. A.3.2.4]
In particular, the fibrant objects in this model structure are the Kan complex-enriched categories, i.e. the strictly ∞-groupoid-enriched ones (see (n,r)-category).
The cofibrations in the Dwyer-Kan-Bergner model structure (Prop. ) are in particular hom-object-wise cofibrations in the classical model structure on simplicial sets, hence hom-object wise monomorphism of simplicial sets:
The Bergner model structure of prop. is a proper model category.
A reference for right properness is (Bergner 04, prop. 3.5). A reference for left properness is (Lurie, A.3.2.4) and (Lurie, A.3.2.25).
(failure of cartesian closed model structure)
While the underlying category $sSet Cat$ of sSet-enriched categories is cartesian closed (on general grounds, see here, also cf. Joyal (2008), §51.2), the construction of $sSet$-enriched product categories and $sSet$-enriched functor categories does not make the Bergner model structure of prop. into a monoidal model category hence not into a cartesian closed model category.
This is in contrast to the model structure on quasi-categories (see there) to which the Bergner structure is Quillen equivalent (see below).
The canonical inclusion functor
of the category of sSet-enriched groupoids (aka Dwyer-Kan simplicial 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 Bergner-model structure (Prop. ) and the Dwyer-Kan model structure on simplicial groupoids
hence we have a Quillen adjunction:
(see also Minichiello, Rivera & Zeinalian (2023), Prop. 2.8)
The operations of forming homotopy coherent nerves, $N$, and of rigidification of quasi-categories, $\mathfrak{C}$, constitute a Quillen equivalence between the Bergner model structure on $sSet Cat$ (Prop. ) and the model structure for quasi-categories.
For more see at relation between sSet-enriched categories and quasi-categories.
The entries of the following table display model categories and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).
general pattern | ||||
---|---|---|---|---|
strict enrichment | (∞,1)-category/(∞,1)-operad | |||
$\downarrow$ | $\downarrow$ | |||
enriched (∞,1)-category | $\hookrightarrow$ | internal (∞,1)-category | ||
(∞,1)Cat | ||||
SimplicialCategories | $-$homotopy coherent nerve$\to$ | SimplicialSets/quasi-categories | RelativeSimplicialSets | |
$\downarrow$simplicial nerve | $\downarrow$ | |||
SegalCategories | $\hookrightarrow$ | CompleteSegalSpaces | ||
(∞,1)Operad | ||||
SimplicialOperads | $-$homotopy coherent dendroidal nerve$\to$ | DendroidalSets | RelativeDendroidalSets | |
$\downarrow$dendroidal nerve | $\downarrow$ | |||
SegalOperads | $\hookrightarrow$ | DendroidalCompleteSegalSpaces | ||
$\mathcal{O}$Mon(∞,1)Cat | ||||
DendroidalCartesianFibrations |
A model category structure on the category of $sSet$-categories with a fixed collection of objects was first given in:
The notion of what is now called Dwyer-Kan equivalences appears in
Dywer, Spalinski and later Rezk then pointed out that there ought to exist a model category structure on the collection of all $sSet$-categories that models the (∞,1)-category of (∞,1)-categories. This was then constructed in:
Also:
Jacob Lurie, Section A.3.2 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, pdf]
Jacob Lurie, $(\infty,2)$-categories and the Goodwillie calculus, GoodwillieI.pdf
Survey and review:
Julie Bergner, Section 3 of: A survey of $(\infty,1)$-categories (arXiv:math/0610239)
Emily Riehl, Section 16 of: Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)
See also
André Joyal, Notes on quasi-categories (2008) [pdf, pdf]
Aaron Mazel-Gee, Quillen adjunctions induce adjunctions of quasicategories, New York Journal of Mathematics Volume 22 (2016) 57-93 (arXiv:1501.03146, publisher)
Recall the slight but crucial difference between the two notions of “simplicial categories”, the other being an internal category in sSet. But also for this latter concept there is a model category structure which presents (infinity,1)-categories, see
Last revised on May 31, 2023 at 16:07:18. See the history of this page for a list of all contributions to it.