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From the Introduction:

Homotopy style idea is a brand new department of arithmetic that mixes facets of a number of diversified fields in a shocking means. it's in accordance with a lately found connection among homotopy conception and kind concept. It touches on themes as possible far away because the homotopy teams of spheres, the algorithms for kind checking, and the definition of vulnerable ∞-groupoids.

Homotopy sort thought brings new principles into the very origin of arithmetic. at the one hand, there's Voevodsky’s refined and gorgeous 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. however, now we have greater inductive kinds, which supply direct, logical descriptions of a few of the elemental areas and structures of homotopy thought: spheres, cylinders, truncations, localizations, and so forth. either rules are most unlikely to seize at once in classical set-theoretic foundations, but if mixed in homotopy sort thought, they enable a wholly new type of “logic of homotopy types”.

This indicates a brand new notion of foundations of arithmetic, with intrinsic homotopical content material, an “invariant” belief of the gadgets of arithmetic — and handy computing device implementations, which could function a realistic relief to the operating mathematician. this is often the Univalent Foundations program.

The current ebook is meant as a primary systematic exposition of the fundamentals of univalent foundations, and a set of examples of this new kind of reasoning — yet with out requiring the reader to grasp or study any formal good judgment, or to exploit any desktop facts assistant. We think that univalent foundations will ultimately turn into a attainable substitute to set thought because the “implicit origin” for the unformalized arithmetic performed by way of so much mathematicians.

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