Joyal's theorem

The Joyal theorem is a theorem in category theory, this theorem proves the statement "an ∞-groupoid is a Kan complex", which is a version of the Homotopy hypothesis.

Joyal extension theorem

Let $C$ be quasicategory and let $u:X\rightarrow Y$ be a morphism of $C$. The following conditions are equivalent:

(1)The morphism u is an isomorphism.

(2)

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

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

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

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

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

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

Joyal lifting theorem

Let $p:C\rightarrow D$ be an inner fibration between quasicategories, and let $f\in C_{1}$ be an edge such that $p(f)$ is an isomorphism in D. The following are equivalent:

(1) The edge f is an isomorphism in C.

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

admits a lift.

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

admits a lift.

Reference

• 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.
• "Theorem 4.4.2.6 (Joyal)". Kerodon.
• Haugseng, Rune. "Introduction to ∞-Categories" (PDF).