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{{primary sources|date=July 2025}}
{{Short description|Model structure on the category of simplicial sets}}
In [[higher category theory]] in mathematics, the '''Joyal 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 [[∞-category|∞-categories]] and it furthermore models the [[homotopy theory]] of [[CW complex]]es up to [[homotopy equivalence]], with the correspondence between simplicial sets and CW complexes being given by the [[Geometric realization functor|geometric realization]] and the singular functor. The Joyal model structure is named after [[André Joyal]].
== Definition ==
The Joyal model structure is given by:
* Fibrations are
* Cofibrations are [[monomorphism]]s.<ref name=":9">Lurie 2009, ''Higher Topos Theory'', Theorem 1.3.4.1.</ref>
* Weak equivalences are ''weak categorical equivalences'',<ref name=":11">Joyal 2008, Theorem 6.12.</ref> hence morphisms between simplicial sets, whose [[Geometric realization functor|geometric realization]] is a [[homotopy equivalence]] between [[CW complex]]es.
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* Fiberant objects of the Joyal model structure, hence simplicial sets <math>X</math>, for which the [[Terminal object|terminal]] morphism <math>X\xrightarrow{!}\Delta^0</math> is a fibration, are the [[∞-category|∞-categories]].<ref name=":11" /><ref>Lurie 2009, ''Higher Topos Theory'', p. 58 & Theorem 2.3.6.4.</ref><ref name=":0" />
* Cofiberant objects of the Joyal model structure, hence simplicial sets <math>X</math>, for which the [[Initial object|initial]] morphism <math>\emptyset\xrightarrow{!}X</math> is a cofibration, are all simplicial sets.
* The Joyal model structure is [[Left proper model structure|left proper]], which follows directly from all objects being cofibrant.<ref name=":8">Lurie 2009, ''Higher Topos Theory'', Proposition A.2.3.2.</ref> This means that weak categorical equivalences are preversed by [[Pushout (category theory)|pushout]] along its cofibrations (the monomorphisms). The Joyal model structure is '''not''' right proper
\Delta^0+\Delta^0\hookrightarrow\Delta^1</math>, is not due for example the different number of connected components.<ref>Lurie 2009, ''Higher Topos Theory'', Remark 1.3.4.3.</ref> This counterexample doesn't work for the [[Kan–Quillen model structure]] since <math>\Delta^1\cong\{0\rightarrow 2\}\hookrightarrow\Delta^2</math> is not a Kan fibration. But the pullback of weak categorical equivalences along left or right Kan fibrations is again a weak categorical equivalence.<ref>Joyal 2008, Remark 6.13.</ref>
* The Joyal model structure is a [[Cisinski model structure]] and in particular cofibrantly generated. Cofibrations (monomorphisms) are generated by the boundary inclusions <math>\partial\Delta^n\hookrightarrow\Delta^n</math> and acyclic cofibrations (inner anodyne extensions) are generated by inner horn inclusions <math>\Lambda_k^n\hookrightarrow\Delta^n</math> (with <math>n\geq 2</math> and <math>0<k<n</math>).
* Weak categorical equivalences are final.<ref>Cisinski 2019, Proposition 5.3.1.</ref>
* Inner anodyne extensions are weak categorical equivalences.<ref>Joyal 2008, Corollary 2.29. on p. 239</ref><ref>Lurie 2009, ''Higher Topos Theory'', Lemma 1.3.4.2.</ref>
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\mathbf{sSet}_\mathrm{KQ}\rightarrow\mathbf{sSet}_\mathrm{J}</math> preserves both cofibrations and acyclic cofibrations, hence as a left adjoint with the identity <math>\operatorname{Id}\colon
\mathbf{sSet}_\mathrm{J}\rightarrow\mathbf{sSet}_\mathrm{KQ}</math> as right adjoint forms a [[Quillen adjunction]].
== Local weak categorical equivalence ==
For a simplicial set <math>B</math> and a morphism of simplicial sets <math>f\colon X\rightarrow Y</math> over <math>B</math> (so that there are morphisms <math>p\colon X\rightarrow B</math> and <math>q\colon Y\rightarrow B</math> with <math>p=q\circ f</math>), the following conditions are equivalent:<ref name=":1">Cisinski 2019, Lemma 5.3.9.</ref>
* For every <math>n</math>-simplex <math>\sigma\colon\Delta^n\rightarrow B</math>, the induced map <math>\Delta^n\times_B\sigma\colon
\Delta^n\times_BX\rightarrow\Delta^n\times_BY</math> is a weak categorical equivalence.
* For every morphism <math>g\colon
A\rightarrow B</math>, the induced map <math>A\times_Bg\colon
A\times_BX\rightarrow A\times_BY</math> 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.<ref name=":1" />
== Literature ==
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