Download E-books Homotopy Type Theory: Univalent Foundations of Mathematics PDF

By Univalent Foundations Program

From the Introduction:

Homotopy variety idea is a brand new department of arithmetic that mixes points of numerous diversified fields in a stunning method. it really is in keeping with a lately chanced on connection among homotopy conception and sort conception. It touches on themes as likely far away because the homotopy teams of spheres, the algorithms for sort checking, and the definition of susceptible ∞-groupoids.

Homotopy sort concept brings new principles into the very beginning of arithmetic. at the one hand, there's Voevodsky’s refined and lovely univalence axiom. The univalence axiom implies, particularly, that isomorphic buildings may be pointed out, a precept that mathematicians were fortunately utilizing on paintings- days, regardless of its incompatibility with the “official” doctrines of traditional foundations. nevertheless, we've got greater inductive kinds, which supply direct, logical descriptions of a few of the fundamental areas and structures of homotopy thought: spheres, cylinders, truncations, localizations, and so forth. either principles are most unlikely to seize at once in classical set-theoretic foundations, but if mixed in homotopy style concept, they allow a completely new type of “logic of homotopy types”.

This indicates a brand new perception of foundations of arithmetic, with intrinsic homotopical content material, an “invariant” notion of the items of arithmetic — and handy computer implementations, which could function a realistic reduction to the operating mathematician. this can be the Univalent Foundations program.

The current booklet is meant as a primary systematic exposition of the fundamentals of univalent foundations, and a suite of examples of this new sort of reasoning — yet with out requiring the reader to grasp or examine any formal good judgment, or to take advantage of any desktop facts assistant. We think that univalent foundations will finally develop into a practicable replacement to set idea because the “implicit starting place” for the unformalized arithmetic performed by way of so much mathematicians.