model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
Background
Basic concepts
equivalences in/of -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 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 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 -full and faithful functor in that for all objects 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 between model categories induces a Dwyer-Kan-equivalence 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 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 such that
for all the morphism is a fibration in the classical model structure on simplicial sets;
the induced functor 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 of sSet-enriched categories is cartesian closed (on general grounds, see here, also cf. Joyal (2008), §51.2), the construction of -enriched product categories and -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, , and of rigidification of quasi-categories, , constitute a Quillen equivalence between the Bergner model structure on (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 -monoidal (∞,1)-categories (fourth table).
A model category structure on the category of -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 -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, -categories and the Goodwillie calculus, GoodwillieI.pdf
Survey and review:
Julie Bergner, Section 3 of: A survey of -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.