Kan–Quillen model structure

In higher category theory, the Kan–Quillen model structure is a special model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called fibrations, cofibrations and weak equivalences, which fulfill the properties of a model structure. Its fibrant objects are all Kan complexes and it furthermore models the homotopy theory of CW complexes up to weak homotopy equivalence, with the correspondence between simplicial sets, Kan complexes and CW complexes being given by the geometric realization and the singular functor (Milnor's theorem). The Kan–Quillen model structure is named after Daniel Kan and Daniel Quillen.

Definition

The Kan–Quillen model structure is given by:

Fibrations are Kan fibrations.^[1]
Cofibrations are monomorphisms.^[1]^[2]
Weak equivalences are weak homotopy equivalences,^[1] hence morphisms between simplicial sets, whose geometric realization is a weak homotopy equivalence between CW complexes.
Trivial cofibrations are anodyne extensions.^[3]

The category of simplicial sets $\mathbf {sSet}$ with the Kan–Quillen model structure is denoted $\mathbf {sSet} _{\mathrm {KQ} }$ .

Properties

Fiberant objects of the Kan–Quillen model structure, hence simplicial sets $X$ , for which the terminal morphism $X\xrightarrow {!} \Delta ^{0}$ is a fibration, are the Kan complexes.^[1]
Cofiberant objects of the Kan–Quillen model structure, hence simplicial sets $X$ , for which the initial morphism $\emptyset \xrightarrow {!} X$ is a cofibration, are all simplicial sets.
The Kan–Quillen model structure is proper.^[1]^[4] This means that weak homotopy equivalences are both preversed by pullback along its fibrations (Kan fibrations) as well as pushout along its cofibrations (monomorphisms). Left properness follows directly since all objects are cofibrant.^[5]
The Kan–Quillen model structure is cofibrantly generated. Cofibrations (monomorphisms) are generated by the boundary inclusions $\partial \Delta ^{n}\hookrightarrow \Delta ^{n}$ and acyclic cofibrations (anodyne extensions) are generated by horn inclusions $\Lambda _{k}^{n}\hookrightarrow \Delta ^{n}$ .
Weak homotopy equivalences are closed under finite products.^[6]
Since the Joyal model structure also has monomorphisms as cofibrations^[7] and every weak homotopy equivalence is a weak categorical equivalence, the identity $\operatorname {Id} \colon \mathbf {sSet} _{\mathrm {KQ} }\rightarrow \mathbf {sSet} _{\mathrm {J} }$ preserves both cofibrations and acyclic cofibrations, hence as a left adjoint with the identity $\operatorname {Id} \colon \mathbf {sSet} _{\mathrm {J} }\rightarrow \mathbf {sSet} _{\mathrm {KQ} }$ as right adjoint forms a Quillen adjunction.