natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
basic constructions:
strong axioms
further
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An introductory textbook on homotopy type theory and its use as a foundations of mathematics:
Homotopy Type Theory – Univalent Foundations of Mathematics
(web, pdf, PM wiki version)
About the book (from the cover)
Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory offers a new “univalent” foundation of mathematics, in which a central role is played by Voevodsky‘s univalence axiom and higher inductive types. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.
A more detailed description of the contents here:
http://golem.ph.utexas.edu/category/2013/06/the_hott_book.html#more
And a description of the “sociology” of the writing of the book is here:
Andrej Bauer, The HoTT book (web, alternative)
Last revised on January 7, 2016 at 23:35:51. See the history of this page for a list of all contributions to it.