Co- and contravariant model structure

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

• 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.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 $A$. Contravariant fibrations between two such right fibrant objects over $A$ are exactly the right fibrations.Lurie 2009, Remark 2.1.4.12.Cisinski 2019, Theorem 4.1.5

The slice category $\backslash mathbf\{sSet\}/A$ with the co- and contravariant model structure is denoted $(\backslash mathbf\{sSet\}/A)\_\backslash mathrm\{cov\}$ and $(\backslash mathbf\{sSet\}/A)\_\backslash mathrm\{cont\}$ respectively.

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

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:Lurie 2009, Notation 2.2.3.8.Cisinski 2019, 4.4.8. & 4.4.19.

: $\backslash operatorname\{LFib\}(A)$

:=\operatorname{Ho}((\mathbf{sSet}_\mathrm{KQ}/A)_\mathrm{cov})

: $\backslash operatorname\{RFib\}(A)$

:=\operatorname{Ho}((\mathbf{sSet}_\mathrm{KQ}/A)_\mathrm{cont})

Since $\backslash Delta^0$ is the terminal object of $\backslash mathbf\{sSet\}$, one in particular has:Cisinski 2019, Eq. (4.4.21.2)

: $\backslash operatorname\{Ho\}(\backslash mathbf\{sSet\}\_\backslash mathrm\{KQ\})$

=\operatorname{LFib}(\Delta^0)

=\operatorname{RFib}(\Delta^0).

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)

: $\backslash operatorname\{LFib\}(A^\backslash mathrm\{op\})$

=\operatorname{RFib}(A).

Let $p\backslash colon$

A\rightarrow B be a morphism of simplicial sets, then there is a functor $p\_!\backslash colon$

\mathbf{sSet}/A\rightarrow\mathbf{sSet}/B by postcomposition and a functor $p^*\backslash colon$

\mathbf{sSet}/B\rightarrow\mathbf{sSet}/A by pullback with an adjunction $p\_!\backslash dashv\; p^*$. Since the latter commutes with all colimits, it also has a right adjoint $p\_*\backslash colon$

\mathbf{sSet}/A\rightarrow\mathbf{sSet}/B with $p^*\backslash dashv\; p\_*$. 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.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)

: $\backslash mathbf\{L\}p\_!\backslash colon\backslash operatorname\{RFib\}(A)\backslash rightleftarrows\backslash operatorname\{RFib\}(B)\backslash colon\backslash mathbf\{R\}p^*,$

: $\backslash mathbf\{L\}p^*\backslash colon\backslash operatorname\{RFib\}(B)\backslash rightleftarrows\backslash operatorname\{RFib\}(A)\backslash colon\backslash mathbf\{R\}p\_*.$

• For a functor of ∞-categories $p\backslash colon$

A\rightarrow B , the following conditions are equivalent:Cisinski 2019, Proposition 4.5.2.

• $p\backslash colon$

A\rightarrow B is fully faithful.

• $\backslash mathbf\{L\}p\_!\backslash colon$

\operatorname{LFib}(A)\rightarrow\operatorname{LFib}(B) is fully faithful.

• $\backslash mathbf\{L\}p\_!\backslash colon$

\operatorname{RFib}(A)\rightarrow\operatorname{RFib}(B) is fully faithful.

• For an essential surjective functor of ∞-categories $p\backslash colon$

A\rightarrow B , the functor $\backslash mathbf\{R\}p^*\backslash colon$

\operatorname{RFib}(B)\rightarrow\operatorname{RFib}(A) is conservative.Cisinski 2019, Proposition 4.5.5.

• Every equivalence of ∞-categories $p\backslash colon$

A\rightarrow B induces equivalence of categories:Cisinski 2019, Corollary 4.5.6.

• : $\backslash mathbf\{L\}p\_!\backslash colon$

\operatorname{LFib}(A)\rightleftarrows\operatorname{LFib}(B),

• : $\backslash mathbf\{L\}p\_!\backslash colon$

\operatorname{RFib}(A)\rightleftarrows\operatorname{RFib}(B),

• All inner horn inclusions $i\backslash colon$

\Lambda_k^n\hookrightarrow\Delta^n (with $n\backslash geq\; 2$ and

• : $\backslash mathbf\{L\}i\_!\backslash colon\backslash operatorname\{RFib\}(\backslash Lambda\_k^n)\backslash rightarrow\backslash operatorname\{RFib\}(\backslash Delta^n).$

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

• {{cite book |last1=Lurie |first1=Jacob |author-link=Jacob Lurie |title=Higher Topos Theory |title-link=Higher Topos Theory |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14049-0 |series=Annals of Mathematics Studies |volume=170 |mr=2522659 |arxiv=math.CT/0608040}}
• {{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2019-06-30 |publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}

Category:Higher category theory

Category:Simplicial sets