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
A model category structure whose fibrant objects are precisely the Reedy fibrant Segal categories. This is a presentation of the (∞,1)-category (∞,1)Cat from the point of view of (weakly) ∞Grpd-enriched categories.
Write $PreSegalCat \hookrightarrow [\Delta^{op}, sSet]$ for the full subcategory on those bisimplicial sets $X$ for which $X_0$ is a discrete simplicial set (the “precategories”).
The nerve functor
has a left adjoint (“fundamental category” functor)
Say a morphism $f : X \to Y$ in $PreSegalCat$ is
full and faithful if for all $a,b \in X_0$ the induced morphism
is a weak homotopy equivalence of simplicial sets;
essentially surjective if $\tau_1(f)$ is essentially surjective.
a categorical equivalence if it is both full and faithful as well as essentially surjective.
There is an essentially unique completion functor
equipped with a natural transformation
such that for all pre-Segal categories $X$
$compl(X)$ is a Segal category;
$i_X \colon X \to compl(X)$ is an isomorphism on the sets of objects;
$i_X$ is a categorical equivalence if $X$ is already a Segal category;
$compl(i_X)$ is a categorical equivalence.
This is (HS, def. 2.1, lemma 2.2).
Say a morphism $f : X \to Y$ in $PreSegalCat$ is
a cofibration precisely if it is a monomorphism;
a weak equivalence precisely if its completion $compl(f)$ by prop. 1 is a categorical equivalence.
(…)
This defines a model category structure for Segal categories (…)
(…)
It follows that a map $X \to Y$ between Segal categories is a weak equivalence precisely if it is a categorical equivalence.
Because by prop. 1 we have a commuting square of the form
where the horizontal morphisms are categorical equivalences, and by prop. 2 these satisfy 2-out-of-3.
Equipped with the classes of maps defined in def. 2, $PreSegalCat$ is a model category which is
Cartesian closure is shown in (Pellissier). The fact that it is a Cisinski model structure follows from the result in (Bergner).
See table - models for (infinity,1)-categories.
The model structure for Segal categories was introduced in
(even for Segal n-categories). An alternative proof of the existence of this model structure is given in section 5 of
The cartesian closure of the model structure was established in
The fact that the fibrant Segal categories in this model structure are precisely the Reedy fibrant Segal categories is due to
Model structures for Segal categories enriched over more general (∞,1)-categories are discussed in section 2 of