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 ${\displaystyle C}$ 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} .

Joyal lifting theorem

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.