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.

Joyal extension theorem

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

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