π n (S n ) in Homotopy Type Theory

A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory

“…Using these tools, some elementary constructions from algebraic topology or homotopy theory have been developed and formalized using the proof assistants Agda (Norell 2007) and Coq (Coq Development Team 2009). These include calculations of some homotopy groups of spheres (Licata and Shulman 2013;Licata and Brunerie 2013;The Univalent Foundations Program 2013) and some homotopy equivalences (Licata and Brunerie 2015); constructions of the Hopf fibration (The Univalent Foundations Program 2013, §8.5), of covering spaces (Hou (Favonia) 2014), and of Eilenberg-Mac Lane spaces (Licata and Finster 2014); and proofs of the Freudenthal suspension theorem (The Univalent Foundations Program 2013, §8.6), the van Kampen theorem (The Univalent Foundations Program 2013, §8.7), and the Mayer-Vietoris theorem (Cavallo 2014). These developments are interesting from a formalization perspective because using the homotopy structure of types results in short, clean mechanized proofs.…”

“…In developing synthetic homotopy theory, we have found that the same result can often be proved in ways that feel more "typetheoretic" or in ways that feel more "homotopy-theoretic"-for example, when a standard mathematical argument is translated to type theory and viewed from a perspective that emphasizes computation and normal forms and cut-free proofs, it sometimes looks indirect or reducible. The present work is the result of a back-and-forth between category theorists and type theorists, beginning with a calculation of the fundamental group of the circle by Shulman (Shulman 2011b), which was "reduced" by Licata (Licata and Shulman 2013), which led to a "type-theoretic" calculation of the diagonal homotopy groups of spheres by Licata and Brunerie (Licata and Brunerie 2013), which led to a calculation by Lumsdaine of the same using a more classical approach (the Freudenthal suspension theorem) (The Univalent Foundations Program 2013, §8.6), which led to the present result. To illustrate this interplay of type theory and homotopy theory, we will actually describe two different mechanizations of the Blakers-Massey theorem, one that is more direct in type theory, but involves some calculations that are hard to phrase in traditional homotopy-theoretic terms, and one that would be more familiar to a homotopy theorist, but is less direct, in the sense that it makes use of some intermediate types that are eliminated in the first version.…”

Homotopical patch theory

“…Another is higher inductive types [24,25,30], which are a new class of datatypes, specified by constructors not only for points but also for paths. Higher inductive types were originally introduced to permit basic topological spaces such as circles and spheres to be defined in type theory, and have had significant applications in a line of work on using homotopy type theory to give computer-checked proofs in homotopy theory [18,19,22,32].…”

Higher Inductive Types as Homotopy-Initial Algebras

“…Conversely, type theory is used to formalize and verify existing mathematical proofs using proof assistants such as Coq [19] and Agda [16]. Moreover, type-theoretic insights often help us discover novel proofs of known results which are simpler than their homotopy-theoretic versions: the calculation of π n (S n ) ( [11,9]); the Freudenthal Suspension Theorem [20]; the Blakers-Massey Theorem [20], etc.…”