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 be quasicategory and let be a morphism of . The following conditions are equivalent:

(1) The morphism is an isomorphism.

(2) Let and let be a morphism of simplicial sets for which the initial edge

is equal to . Then can be extended to an n-simplex .

(3) Let and let be a morphism of simplicial sets for which the initial edge

is equal to . Then can be extended to an n-simplex .

Joyal's lifting theorem

Let be an inner fibration (Joyal used mid-fibration) between quasicategories, and let be an edge such that is an isomorphism in . The following are equivalent:

(1) The edge is an isomorphism in .

(2) For all , every diagram of the form

admits a lift.

(3) For all , every diagram of the form

admits a lift.

Notes

References

Further reading