Joyal model structure

The Joyal model structure is given by:

• Fibrations are isofibrations.Cisinski 2019, Theorem 3.6.1.
• Cofibrations are monomorphisms.Lurie 2009, Higher Topos Theory, Theorem 1.3.4.1.
• Weak equivalences are weak categorical equivalences,Joyal 2008, Theorem 6.12. hence morphisms between simplicial sets, whose geometric realization is a homotopy equivalence between CW complexes.
• Trivial cofibrations are inner anodyne extensions.

The category of simplicial sets $\backslash mathbf\{sSet\}$ with the Joyal model structure is denoted $\backslash mathbf\{sSet\}\_\backslash mathrm\{J\}$ (or $\backslash mathbf\{sSet\}\_\backslash mathrm\{Joy\}$ for more joy).

• Fiberant objects of the Joyal model structure, hence simplicial sets $X$, for which the terminal morphism $X\backslash xrightarrow\{!\}\backslash Delta^0$ is a fibration, are the ∞-categories.Lurie 2009, Higher Topos Theory, p. 58 & Theorem 2.3.6.4.
• Cofiberant objects of the Joyal model structure, hence simplicial sets $X$, for which the initial morphism $\backslash emptyset\backslash xrightarrow\{!\}X$ is a cofibration, are all simplicial sets.
• The Joyal model structure is left proper, which follows directly from all objects being cofibrant.Lurie 2009, Higher Topos Theory, Proposition A.2.3.2. This means that weak categorical equivalences are preversed by pushout along its cofibrations (the monomorphisms). The Joyal model structure is not right proper. For example the inclusion $\backslash Lambda\_1^2\backslash hookrightarrow\backslash Delta^2$ is a weak categorical equivalence, but its pullback along the isofibration $\backslash Delta^1\backslash cong\backslash \{0\backslash rightarrow\; 2\backslash \}\backslash hookrightarrow\backslash Delta^2$, which is $(0,1)\backslash colon$

\Delta^0+\Delta^0\hookrightarrow\Delta^1, is not due for example the different number of connected components.Lurie 2009, Higher Topos Theory, Remark 1.3.4.3. This counterexample doesn't work for the Kan–Quillen model structure since $\backslash Delta^1\backslash cong\backslash \{0\backslash rightarrow\; 2\backslash \}\backslash hookrightarrow\backslash Delta^2$ is not a Kan fibration. But the pullback of weak categorical equivalences along left or right Kan fibrations is again a weak categorical equivalence.Joyal 2008, Remark 6.13.

• The Joyal model structure is a Cisinski model structure and in particular cofibrantly generated. Cofibrations (monomorphisms) are generated by the boundary inclusions $\backslash partial\backslash Delta^n\backslash hookrightarrow\backslash Delta^n$ and acyclic cofibrations (inner anodyne extensions) are generated by inner horn inclusions $\backslash Lambda\_k^n\backslash hookrightarrow\backslash Delta^n$ (with $n\backslash geq\; 2$ and
• Weak categorical equivalences are final.Cisinski 2019, Proposition 5.3.1.
• Inner anodyne extensions are weak categorical equivalences.Joyal 2008, Corollary 2.29. on p. 239Lurie 2009, Higher Topos Theory, Lemma 1.3.4.2.
• Weak categorical equivalences are closed under finite productsJoyal 2008, Proposition 2.28. on p. 239Lurie 2009, Higher Topos Theory, Corollary 1.3.4.4.Cisinski 2019, Corollary 3.6.3. and small filtered colimits.Joyal 2008, Corollary 6.10. on p. 299Cisinski 2019, Corollary 3.9.8.
• Since the Kan–Quillen model structure also has monomorphisms as cofibrationsCisinski 2019, Theorem 3.1.8. and every weak homotopy equivalence is a weak categorical equivalence,Joyal 2008, Corollary 6.16. on p. 301 the identity $\backslash operatorname\{Id\}\backslash colon$

\mathbf{sSet}_\mathrm{KQ}\rightarrow\mathbf{sSet}_\mathrm{J} preserves both cofibrations and acyclic cofibrations, hence as a left adjoint with the identity $\backslash operatorname\{Id\}\backslash colon$

\mathbf{sSet}_\mathrm{J}\rightarrow\mathbf{sSet}_\mathrm{KQ} as right adjoint forms a Quillen adjunction.

For a simplicial set $B$ and a morphism of simplicial sets $f\backslash colon\; X\backslash rightarrow\; Y$ over $B$ (so that there are morphisms $p\backslash colon\; X\backslash rightarrow\; B$ and $q\backslash colon\; Y\backslash rightarrow\; B$ with $p=q\backslash circ\; f$), the following conditions are equivalent:Cisinski 2019, Lemma 5.3.9.

• For every $n$-simplex $\backslash sigma\backslash colon\backslash Delta^n\backslash rightarrow\; B$, the induced map $\backslash Delta^n\backslash times\_B\backslash sigma\backslash colon$

\Delta^n\times_BX\rightarrow\Delta^n\times_BY is a weak categorical equivalence.

• For every morphism $g\backslash colon$

A\rightarrow B, the induced map $A\backslash times\_Bg\backslash colon$

A\times_BX\rightarrow A\times_BY is a weak categorical equivalence.

Such a morphism is called a local weak categorical equivalence.

• Every local weak categorical equivalence is a weak categorical equivalence.

• {{cite web |last=Joyal |first=André |author-link=André Joyal |date=2008 |title=The Theory of Quasi-Categories and its Applications |url=https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf |language=en}}
• {{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=}}

• model structure on simplicial sets at the nLab
• [https://kerodon.net/tag/01GP The Homotopy Theory of ∞-Categories] at Kerodon

Category:Higher category theory

Category:Homotopy theory

Category:Simplicial sets