Joyal's extension and lifting theorems

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.

The theorem was introduced by André Joyal.

Joyal's extension theorem

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

(1) The morphism $u$ is an isomorphism.

(2) Let $n\ge 2$ and let ${\sigma }_0:{\mathrm{\Lambda }}_0^n\to C$ be a morphism of simplicial sets for which the initial edge

${\mathrm{\Delta }}^1\cong N_{\bullet }(\{0<1\})\to {\mathrm{\Lambda }}_0^n\stackrel{{\sigma }_0}{\to }C$

is equal to $u$. Then ${\sigma }_0$ can be extended to an n-simplex $\sigma :{\mathrm{\Delta }}^n\to C$.

(3) Let $n\ge 2$ and let ${\sigma }_0:{\mathrm{\Lambda }}_n^n\to C$ be a morphism of simplicial sets for which the initial edge

${\mathrm{\Delta }}^1\cong N_{\bullet }(\{n-1<n\})\to {\mathrm{\Lambda }}_n^n\stackrel{{\sigma }_0}{\to }C$

is equal to $u$. Then ${\sigma }_0$ can be extended to an n-simplex $\sigma :{\mathrm{\Delta }}^n\to C$.

Joyal's lifting theorem

Let $p:C\to D$ be an inner fibration (Joyal used mid-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 $n\ge 2$, every diagram of the form

admits a lift.

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

admits a lift.

References

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Further reading

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