Co- and contravariant model structure

In higher category theory in mathematics, co- and contravariant model structures are special model structures on slice categories of the category of simplicial sets. On them, postcomposition and pullbacks (due to its application in algebraic geometry also known as base change) induce adjoint functors, which with the model structures can even become Quillen adjunctions.

Definition Let be a simplicial set, then there is a slice category . With the choice of a model structure on , for example the Joyal or Kan–Quillen model structure, it induces a model structure on .

Covariant cofibrations are monomorphisms. Covariant fibrant objects are the left fibrant objects over . Covariant fibrations between two such left fibrant objects over are exactly the left fibrations.Lurie 2009, Definition 2.1.4.5.Cisinski 2019, Theorem 4.4.14 Contravariant cofibrations are monomorphisms. Contravariant fibrant objects are the right fibrant objects over . Contravariant fibrations between two such right fibrant objects over are exactly the right fibrations.Lurie 2009, Remark 2.1.4.12.Cisinski 2019, Theorem 4.1.5

The slice category with the co- and contravariant model structure is denoted and respectively.

Properties

The covariant model structure is left proper and combinatorical.Lurie 2009, Proposition 2.1.4.7.

Homotopy categories For any model category, there is a homotopy category associated to it by formally inverting all weak equivalences. In homotopical algebra, the co- and contravariant model structures of the Kan–Quillen model structure with weak homotopy equivalences as weak equivalences are of particular interest. For a simplicial set , let:Lurie 2009, Notation 2.2.3.8.Cisinski 2019, 4.4.8. & 4.4.19.

Since is the terminal object of , one in particular has:Cisinski 2019, Eq. (4.4.21.2)

Since the functor of the opposite simplicial set is a Quillen equivalence between the co- and contravariant model structure, one has:Cisinski 2019, Eq (4.4.19.1)

Quillen adjunctions Let be a morphism of simplicial sets, then there is a functor by postcomposition and a functor by pullback with an adjunction . Since the latter commutes with all colimits, it also has a right adjoint with . For the contravariant model structure (of the Kan–Quillen model structure), the former adjunction is always a Quillen adjunction, while the latter is for proper.Cisinski 2019, Proposition 4.4.6. & Proposition 4.4.7. This results in derived adjunctions:Cisinski 2019, Equation (4.4.8.2) & Equation (4.4.8.3)

Properties

For a functor of ∞-categories , the following conditions are equivalent:Cisinski 2019, Proposition 4.5.2. is fully faithful. is fully faithful. is fully faithful. For an essential surjective functor of ∞-categories , the functor is conservative.Cisinski 2019, Proposition 4.5.5. Every equivalence of ∞-categories induces equivalence of categories:Cisinski 2019, Corollary 4.5.6.

All inner horn inclusions (with and ) induce an equivalence of categories:Cisinski 2019, Proposition 5.2.1.

Co- and contravariant model structure

Bron en licentie