Constructive homotopy theory of marked semisimplicial sets

Abstract

We develop the homotopy theory of semisimplicial sets constructively and without reference to point-set topology to obtain a constructive model for $\omega$-groupoids. Most of the development is folklore, but for a few results the author is unaware of previously known constructive proofs. These include the statements that the unit of the free simplicial set adjunction is valued in weak equivalences and that the geometric product and cartesian product of fibrant semisimplicial sets are weakly equivalent. We then extend the development to marked semisimplicial sets in order to obtain a constructive model for $(\omega, 1)$-categories.