1 research outputs found
Homotopy type theory and Voevodsky's univalent foundations
Recent discoveries have been made connecting abstract homotopy theory and the
field of type theory from logic and theoretical computer science. This has
given rise to a new field, which has been christened "homotopy type theory". In
this direction, Vladimir Voevodsky observed that it is possible to model type
theory using simplicial sets and that this model satisfies an additional
property, called the Univalence Axiom, which has a number of striking
consequences. He has subsequently advocated a program, which he calls univalent
foundations, of developing mathematics in the setting of type theory with the
Univalence Axiom and possibly other additional axioms motivated by the
simplicial set model. Because type theory possesses good computational
properties, this program can be carried out in a computer proof assistant. In
this paper we give an introduction to homotopy type theory in Voevodsky's
setting, paying attention to both theoretical and practical issues. In
particular, the paper serves as an introduction to both the general ideas of
homotopy type theory as well as to some of the concrete details of Voevodsky's
work using the well-known proof assistant Coq. The paper is written for a
general audience of mathematicians with basic knowledge of algebraic topology;
the paper does not assume any preliminary knowledge of type theory, logic, or
computer science.Comment: 48 pages, 14 figure