By Univalent Foundations Program

From the Introduction:

Homotopy variety thought is a brand new department of arithmetic that mixes points of a number of assorted fields in a stunning method. it really is in response to a lately stumbled on connection among homotopy conception and kind concept. It touches on themes as possible far-off because the homotopy teams of spheres, the algorithms for style checking, and the definition of vulnerable ∞-groupoids.

Homotopy variety idea brings new principles into the very starting place of arithmetic. at the one hand, there's Voevodsky’s sophisticated and lovely univalence axiom. The univalence axiom implies, particularly, that isomorphic constructions 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, we've got greater inductive forms, which supply direct, logical descriptions of a few of the fundamental areas and structures of homotopy concept: spheres, cylinders, truncations, localizations, and so on. either rules are very unlikely to catch at once in classical set-theoretic foundations, but if mixed in homotopy kind idea, they enable a completely new type of “logic of homotopy types”.

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

The current publication is meant as a primary systematic exposition of the fundamentals of univalent foundations, and a set of examples of this new form of reasoning — yet with no requiring the reader to grasp or research any formal good judgment, or to take advantage of any laptop facts assistant. We think that univalent foundations will finally develop into a possible replacement to set concept because the “implicit origin” for the unformalized arithmetic performed by means of so much mathematicians.

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**Example text**

17 of coherence and strictness remain to be addressed — and doing so will undoubtedly lead to further insights into both concepts. But by far the largest field of work to be done is in the ongoing formalization of everyday mathematics in this new system. Recent successes in formalizing some facts from basic homotopy theory and category theory have been encouraging; some of these are described in Chapters 8 and 9. Obviously, however, much work remains to be done. org, as well as a discussion email list.

Moreover, the type ∏( x:A) R( a, pr1 ( g( x ))) is the result of substituting the function λx. pr1 ( g( x )) for f in the family being summed over in the codomain of ac: ∏( x:A) R( x, pr1 ( f ( x ))) ≡ λ f . ∏( x:A) R( x, f ( x )) (λx. pr1 ( g( x ))). Thus, we have λx. pr1 ( g( x )), λx. pr2 ( g( x )) : ∑( f :A→ B) ∏( x:A) R( x, f ( x )) as required. If we read Π as “for all” and Σ as “there exists”, then the type of the function ac expresses: if for all x : A there is a y : B such that R( x, y), then there is a function f : A → B such that for all x : A we have R( x, f ( x )).

It is clear that we could construct this type out of coproduct and unit types as 1 + 1. However, since it is used frequently, we give the explicit rules here. Indeed, we are going to observe that we can also go the other way and derive binary coproducts from Σ-types and 2. To derive a function f : 2 → C we need c0 , c1 : C and add the defining equations f (02 ) : ≡ c 0 , f (12 ) : ≡ c 1 . The recursor corresponds to the if-then-else construct in functional programming: rec2 : ∏ C → C → 2 → C C:U with the defining equations rec2 (C, c0 , c1 , 02 ) :≡ c0 , rec2 (C, c0 , c1 , 12 ) :≡ c1 .