We prove the conjecture that any Grothendieck $(\infty, 1) $-topos can be presented by a
Quillen model category that interprets homotopy type theory with strict univalent universes …

The goal of this thesis is to prove that $\pi_4 (S^ 3)\simeq\mathbb {Z}/2\mathbb {Z} $ in
homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof …

Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by
category theory and homotopy theory. With these extensions, type theory can be used to …

Homotopy Type Theory is a new field of mathematics based on the recently-discovered
correspondence between Martin-Löf's constructive type theory and abstract homotopy …

Homotopy theory can be developed synthetically in homotopy type theory, using types to
describe spaces, the identity type to describe paths in a space, and iterated identity types to …

We generalize the adjoint logics of Benton and Wadler [1994, 1996] and Reed [2009] to
allow multiple different adjunctions between the same categories. This provides insight into …

We study different formalizations of finite sets in homotopy type theory to obtain a general
definition that exhibits both the computational facilities and the proof principles expected …

We present a development of cellular cohomology in homotopy type theory. Cohomology
associates to each space a sequence of abelian groups capturing part of its structure, and …

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence
with homotopy theory and higher category theory. In homotopy type theory, the propositional …

The study of equality types is central to homotopy type theory. Characterizing these types is
often tricky, and various strategies, such as the encode-decode method, have been …