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This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book New Spaces for Mathematics and Physics (ed. Gabriel Catren and Mathieu Anel).
Univalent homotopy type theory (HoTT) may be seen as a language for the category of $infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of fac
We propose foundations for a synthetic theory of $(infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary t
We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and significant amounts of synthetic
We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of adjoint logic in which the discretization and codiscretization modalities are characterized using a judgmental formalism of crisp variables. T
We show that the classifying topos for the theory of fields does not satisfy De Morgans law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.