on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
equivalences in/of $(\infty,1)$-categories
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:
there is a model structure on $SSet$-enriched categories whose fibrant objects are ∞-groupoid/Kan complex-enriched categories and which models (∞,1)-categories;
there is another model structure on sSet-categories whose fibrant objects are (∞,1)-category/quasi-category-enriched categories, and which model (∞,2)-categories.
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.
Definition 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.
Such a morphism is also called a Dwyer-Kan weak equivalence after the work by Dwyer-Kan on simplicial localization.
Proposition A Quillen equivalence $C \stackrel{\leftarrow}{\to} D$ between model categories induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ between their simplicial localizations.
Proposition The category SSet Cat of small simplicially enriched categories carries the structure of a model category with
weak equivalences the Dwyer-Kan equivalences;
fibrations those sSet-enriched functors $F : C \to D$ such that
for all $x, y \in C$ the morphism $F_{x,y} : C(x,y) \to D(F(x), F(y))$ is a fibration in the standard model structure on simplicial sets;
the induced functor $\pi_0(F) : Ho(C) \to Ho(D)$ on homotopy categories is an isofibration.
In particular, the fibrant objects in this structure are the Kan complex-enriched categories, i.e. the strictly ∞-groupoid-enriched ones (see (n,r)-category).
The entries of the following table display models, 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 set of objects was first given 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
A survey is in section 3 of
See also section A.3.2 of
for the moment see (∞,2)-category for more on this
Section A.3 of
Also
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 sturcture which presents (infinity,1)-categories, see