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
If $V$ is a monoidal model category, then in many cases there is a model category of $V$-enriched categories. This includes the model structure on simplicial categories and the model structure on dg-categories, for instance.
Let $V$ be a monoidal model category. The localization functor $\gamma: V \to Ho(V)$ is then a lax monoidal functor, and hence any $V$-category $C$ induces a $Ho(V)$-category $\gamma_\bullet C$. The homotopy category of a $V$-category $C$ is the underlying ordinary category $(\gamma_\bullet C)_o$. We say a $V$-functor $F:C\to D$ is locally X if each morphism $F:C(x,y) \to D(F x, F y)$ is X.
Define a $V$-functor $F:C\to D$ to be:
A weak equivalence if $\gamma_\bullet F :\gamma_\bullet C \to \gamma_\bullet D$ is an equivalence of $Ho(V)$-categories (that is, an internal equivalence in the 2-category of $Ho(V)$-categories). This is equivalent to asking that (1) $F$ is locally a weak equivalence, and (2) the ordinary functor $(\gamma_\bullet F)_o$ is essentially surjective.
A naive fibration if (1) $F$ is locally a fibration, and (2) $\gamma_\bullet F$ is an isofibration.
Define a $V$-category $C$ to be
By a theorem of Joyal, these weak equivalences and fibrant objects determine at most one model structure on the category $V Cat$. When it exists, it is called the (canonical, categorical) model structure on $V$-categories.
Usually, the fibrations between fibrant objects in this model structure are precisely the naive fibrations (although between non-fibrant objects, the two classes are distinct). Usually also, the trivial fibrations are precisely the weak equivalences that are also naive fibrations, which is to say the $V$-functors that are (1) locally trivial fibrations and (2) surjective on objects.
See the references for general conditions under which this model structure exists.
(Muro, [Muro], Theorem 1.1.) For any combinatorial closed symmetric monoidal model category $V$ satisfying the monoid axiom, the category $Cat(V)$ admits the Dwyer–Kan model structure. Moreover, this model structure is combinatorial.
The model structure on simplicial categories which presents (∞,1)-categories is induced from the Quillen model structure on simplicial sets.
The model structure on simplicial categories which presents (∞,2)-categories is induced from the Joyal model structure on simplicial sets
The model structure on dg-categories is induced from the projective model structure on chain complexes.
The canonical model structure on Cat is induced from the trivial model structure on Set.
The canonical (Lack) model structure on $2 Cat$ is induced from the canonical model structure on $Cat$.
The canonical (Lack) model structure on Gray-categories is induced from the canonical model structure on $2 Cat$.
Jacob Lurie, Higher Topos Theory section A.3.2
Alexandru Stanculescu, “Constructing model categories with prescribed fibrant objects” (arXiv:1208.6005)
Clemens Berger, Ieke Moerdijk, On the homotopy theory of enriched categories (arXiv:1201.2134)
Fernando Muro, Dwyer–Kan homotopy theory of enriched categories. arXiv:1201.1575.
Last revised on January 30, 2021 at 16:09:01. See the history of this page for a list of all contributions to it.