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In mathematics, '''Joyal's theorem''' is a theorem in [[homotopy theory]] that provides necessary and sufficient conditions for the solvability of a certain lifting problem involving [[simplicial set]]s. In particular, in [[higher category theory]], it proves the statement "an [[∞-groupoid]] is a [[Kan complex]]", which is a version of the [[homotopy hypothesis]].<ref>{{harvnb|Cisinski|2023|loc=Theorem 3.5.1.}}</ref>

The theorem was introduced by [[André Joyal]].

==Joyal extension theorem==
Let <math>C</math> be [[quasicategory]] and let <math>u:X \rightarrow Y</math> be a [[morphism]] of <math>C</math>. The following conditions are equivalent:<ref>{{harvnb|Theorem 4.4.2.6 in Kerodon}}</ref><ref>{{harvnb|Rezk|2022||loc=34.2. Theorem}}</ref><ref>{{harvnb|Lurie|2009||loc=Proposition 1.2.4.3}}</ref><ref>{{harvnb|Joyal|2002|loc=Theorem 1.3}}</ref>

(1)The morphism <math>u</math> is an [[isomorphism]].

(2)Let <math>n \geq 2</math> and let <math>\sigma_0 : \Lambda^n_0 \to C</math> be a morphism of simplicial sets for which the initial edge

<math>\Delta^1 \cong N_{\bullet }( \{ 0 < 1\} ) \rightarrow \Lambda^n_ n \xrightarrow {\sigma _0} C</math>

is equal to <math>u</math>. Then <math>\sigma_0</math> can be extended to an [[Kan fibration#Definition of the standard n-simplex|''n''-simplex]] <math>\sigma : \Delta^n \to C</math>.

(3)
Let <math>n \geq 2</math> and let <math>\sigma_0 : \Lambda^n_n \rightarrow C</math> be a morphism of simplicial sets for which the initial edge

<math>\Delta^1 \cong N_{\bullet }( \{ n-1 < n\} ) \rightarrow \Lambda^n_ n \xrightarrow {\sigma _0} C</math>

is equal to <math>u</math>. Then <math>\sigma_0</math> can be extended to an ''n''-simplex <math>\sigma : \Delta^n \to C</math>.

==Joyal lifting theorem==
Let <math>p: C \rightarrow D</math> be an [[inner fibration]] (Joyal used mid-fibration<ref>{{harvnb|Lurie|2009||page=xiv}}</ref>) between quasicategories, and let <math>f  \in C_1</math> be an edge such that <math>p (f)</math> is an isomorphism in <math>D</math>. The following are equivalent:<ref>{{harvnb|Rezk|2022|loc=34.17. Theorem (Joyal lifting).}}</ref><ref>{{harvnb|Haugseng|loc=Theorem 5.3.1.}}</ref><ref>{{harvnb|Kapulkin|Voevodsky|2020|loc=Theorem 2.10}}</ref><ref>{{harvnb|Land|2021|loc=Theorem. 2.1.8}}</ref><ref>{{harvnb|Joyal|2002|loc=Theorem 2.2}}</ref><ref>{{harvnb|Joyal|2008|loc=Theorem 6.13}}</ref>

(1) The edge <math>f</math> is an isomorphism in <math>C</math>.

(2) For all <math>n \geq 2</math>, every diagram of the form

[[File:Joyal lifting theorem.svg]]

admits a [[Kan fibration#Definition of a Kan fibration|lift]].

(3)For all <math>n \geq 2</math>, every diagram of the form

[[File:Joyal lifting theorem 2.svg]]

admits a lift.

==Notes==
<references />

==References==
*{{cite journal |doi=10.1016/S0022-4049(02)00135-4 |title=Quasi-categories and Kan complexes |date=2002 |last1=Joyal |first1=A. |journal=Journal of Pure and Applied Algebra |volume=175 |issue=1–3 |pages=207–222 }}
*{{cite web |last1=Rezk |first1=Charles |title=Introduction to quasicategories |url=https://ncatlab.org/nlab/files/Rezk-IntroToQuasicategories.pdf|date=2022|via=ncatlab.org}}
*{{cite book |doi=10.1007/978-3-030-61524-6_2 |chapter=Joyal's Theorem, Applications, and Dwyer–Kan Localizations|chapter-url={{Google books|1sMqEAAAQBAJ|page=99|plainurl=yes}} |title=Introduction to Infinity-Categories |series=Compact Textbooks in Mathematics |date=2021 |last1=Land |first1=Markus |pages=97–161 |isbn=978-3-030-61523-9|zbl= 1471.18001 }}
*{{cite journal |doi=10.1112/topo.12173 |title=A cubical approach to straightening |date=2020 |last1=Kapulkin |first1=Krzysztof |last2=Voevodsky |first2=Vladimir |journal=Journal of Topology |volume=13 |issue=4 |pages=1682–1700 }}
*{{Cite web |title=Theorem 4.4.2.6 (Joyal) |url=https://kerodon.net/tag/019F |website=Kerodon|ref={{harvid|Theorem 4.4.2.6 in Kerodon}}}}
*{{Cite web |title=Proposition 4.4.2.13. |url=https://kerodon.net/tag/01H0 |website=Kerodon|ref={{harvid|Proposition 4.4.2.13. in Kerodon}}}}
*{{cite web |last1=Haugseng |first1=Rune |title=Introduction to ∞-Categories |url=https://people.math.rochester.edu/faculty/doug/otherpapers/Haugseng-intro.pdf}}
* {{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=2023|publisher=[[Cambridge University Press]] |isbn=978-1108473200 |location= |language=en |authorlink=}}
*{{cite book |last=Lurie |first=Jacob|author-link=Jacob Lurie|date=2009|title=Higher Topos Theory|arxiv=math/0608040|publisher=Princeton University Press|isbn=978-0-691-14048-3}}
*{{cite web |first1=André |last1=Joyal |title=THE THEORY OF QUASI-CATEGORIES (Vol I) Draft version |url=https://math.uchicago.edu/~may/PEOPLE/JOYAL/0newqcategories.pdf |date=2008}}

==Further reading==
*{{nlab|id=extension|title=extension}}
*{{nlab|id=lift|title=lift}}
*{{nlab|id=filler|title=filler}}

[[Category:Theorems in homotopy theory]]