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

• Covariant cofibrations are monomorphisms. Covariant fibrant objects are the left fibrant objects over $A$. Covariant fibrations between two such left fibrant objects over $A$ are exactly the left fibrations.
• Contravariant cofibrations are monomorphisms. Contravariant fibrant objects are the right fibrant objects over $A$. Contravariant fibrations between two such right fibrant objects over $A$ are exactly the right fibrations.

The slice category $𝐬𝐒𝐞𝐭/A$ with the co- and contravariant model structure is denoted $(𝐬𝐒𝐞𝐭/A{)}_{\mathrm{c}\mathrm{o}\mathrm{v}}$ and $(𝐬𝐒𝐞𝐭/A{)}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}$ respectively.

Properties

• The covariant model structure is left proper and combinatorical.

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 $A$, let:

$\mathrm{LFib}(A):=\mathrm{Ho}(({𝐬𝐒𝐞𝐭}_{\mathrm{K}\mathrm{Q}}/A{)}_{\mathrm{c}\mathrm{o}\mathrm{v}})$

$\mathrm{RFib}(A):=\mathrm{Ho}(({𝐬𝐒𝐞𝐭}_{\mathrm{K}\mathrm{Q}}/A{)}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}})$

Since ${\mathrm{\Delta }}^0$ is the terminal object of $𝐬𝐒𝐞𝐭$, one in particular has:

$\mathrm{Ho}({𝐬𝐒𝐞𝐭}_{\mathrm{K}\mathrm{Q}})=\mathrm{LFib}({\mathrm{\Delta }}^0)=\mathrm{RFib}({\mathrm{\Delta }}^0).$

Since the functor of the opposite simplicial set is a Quillen equivalence between the co- and contravariant model structure, one has:

$\mathrm{LFib}(A^{\mathrm{o}\mathrm{p}})=\mathrm{RFib}(A).$

Quillen adjunctions

Let $p:A\to B$ be a morphism of simplicial sets, then there is a functor $p_{!}:𝐬𝐒𝐞𝐭/A\to 𝐬𝐒𝐞𝐭/B$ by postcomposition and a functor $p^{\ast }:𝐬𝐒𝐞𝐭/B\to 𝐬𝐒𝐞𝐭/A$ by pullback with an adjunction $p_{!}⊣p^{\ast }$. Since the latter commutes with all colimits, it also has a right adjoint $p_{\ast }:𝐬𝐒𝐞𝐭/A\to 𝐬𝐒𝐞𝐭/B$ with $p^{\ast }⊣p_{\ast }$. For the contravariant model structure (of the Kan–Quillen model structure), the former adjunction is always a Quillen adjunction, while the latter is for $p$ proper. This results in derived adjunctions:

$𝐋p_{!}:\mathrm{RFib}(A)\rightleftarrows \mathrm{RFib}(B):𝐑p^{\ast },$

$𝐋p^{\ast }:\mathrm{RFib}(B)\rightleftarrows \mathrm{RFib}(A):𝐑p_{\ast }.$

Properties

• For a functor of ∞-categories $p:A\to B$ , the following conditions are equivalent:
  • $p:A\to B$ is fully faithful.
  • $𝐋p_{!}:\mathrm{LFib}(A)\to \mathrm{LFib}(B)$ is fully faithful.
  • $𝐋p_{!}:\mathrm{RFib}(A)\to \mathrm{RFib}(B)$ is fully faithful.
• For an essential surjective functor of ∞-categories $p:A\to B$ , the functor $𝐑p^{\ast }:\mathrm{RFib}(B)\to \mathrm{RFib}(A)$ is conservative.
• Every equivalence of ∞-categories $p:A\to B$ induces equivalence of categories:

  $𝐋p_{!}:\mathrm{LFib}(A)\rightleftarrows \mathrm{LFib}(B),$

  $𝐋p_{!}:\mathrm{RFib}(A)\rightleftarrows \mathrm{RFib}(B),$

• All inner horn inclusions $i:{\mathrm{\Lambda }}_k^n\hookrightarrow {\mathrm{\Delta }}^n$ (with $n\ge 2$ and $0<k<n$) induce an equivalence of categories:

  $𝐋i_{!}:\mathrm{RFib}({\mathrm{\Lambda }}_k^n)\to \mathrm{RFib}({\mathrm{\Delta }}^n).$

See also

• Injective and projective model structure, induced model structures on functor categories

Literature

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