Joyal's theorem

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 sets. In particular, in higher category theory, it proves the statement "an ∞-groupoid is a Kan complex", which is a version of the homotopy hypothesis.^[1]

The theorem was introduced by André Joyal.

Let ${\displaystyle C}$ be quasicategory and let ${\displaystyle u:X\rightarrow Y}$ be a morphism of ${\displaystyle C}$. The following conditions are equivalent:^[2][3][4][5]

(1) The morphism ${\displaystyle u}$ is an isomorphism.

(2) Let ${\displaystyle n\geq 2}$ and let ${\displaystyle \sigma _{0}:\Lambda _{0}^{n}\to C}$ be a morphism of simplicial sets for which the initial edge

${\displaystyle \Delta ^{1}\cong N_{\bullet }(\{0<1\})\rightarrow \Lambda _{0}^{n}\xrightarrow {\sigma _{0}} C}$

is equal to ${\displaystyle u}$. Then ${\displaystyle \sigma _{0}}$ can be extended to an n-simplex ${\displaystyle \sigma :\Delta ^{n}\to C}$.

(3) Let ${\displaystyle n\geq 2}$ and let ${\displaystyle \sigma _{0}:\Lambda _{n}^{n}\rightarrow C}$ be a morphism of simplicial sets for which the initial edge

${\displaystyle \Delta ^{1}\cong N_{\bullet }(\{n-1<n\})\rightarrow \Lambda _{n}^{n}\xrightarrow {\sigma _{0}} C}$

is equal to ${\displaystyle u}$. Then ${\displaystyle \sigma _{0}}$ can be extended to an n-simplex ${\displaystyle \sigma :\Delta ^{n}\to C}$.

Let ${\displaystyle p:C\rightarrow D}$ be an inner fibration (Joyal used mid-fibration^[6]) between quasicategories, and let ${\displaystyle f\in C_{1}}$ be an edge such that ${\displaystyle p(f)}$ is an isomorphism in ${\displaystyle D}$. The following are equivalent:^{[7][8][9][10][11][12]}

(1) The edge ${\displaystyle f}$ is an isomorphism in ${\displaystyle C}$.

(2) For all ${\displaystyle n\geq 2}$, every diagram of the form

admits a lift.

(3) For all ${\displaystyle n\geq 2}$, every diagram of the form

admits a lift.

• Joyal, A. (2002). "Quasi-categories and Kan complexes". Journal of Pure and Applied Algebra. 175 (1–3): 207–222. doi:10.1016/S0022-4049(02)00135-4.
• Rezk, Charles (2022). "Introduction to quasicategories" (PDF) – via ncatlab.org.
• Land, Markus (2021). "Joyal's Theorem, Applications, and Dwyer–Kan Localizations". Introduction to Infinity-Categories. Compact Textbooks in Mathematics. pp. 97–161. doi:10.1007/978-3-030-61524-6_2. ISBN 978-3-030-61523-9. Zbl 1471.18001.
• Kapulkin, Krzysztof; Voevodsky, Vladimir (2020). "A cubical approach to straightening". Journal of Topology. 13 (4): 1682–1700. doi:10.1112/topo.12173.
• "Theorem 4.4.2.6 (Joyal)". Kerodon.
• "Proposition 4.4.2.13". Kerodon.
• Haugseng, Rune. "Introduction to ∞-Categories" (PDF).
• Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
• Lurie, Jacob (2009). Higher Topos Theory. Princeton University Press. arXiv:math/0608040. ISBN 978-0-691-14048-3.
• Joyal, André (2008). "THE THEORY OF QUASI-CATEGORIES (Vol I) Draft version" (PDF).

• extension at the nLab
• lift at the nLab
• filler at the nLab