Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHomotopy type theory: the logic of space
96
0
0.0
(
0
)
Authors
Michael Shulman
Ask ChatGPT about the research
No Arabic abstract
This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book New Spaces for Mathematics and Physics (ed. Gabriel Catren and Mathieu Anel).
rate research
Read More
Univalent homotopy type theory (HoTT) may be seen as a language for the category of $infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a localization higher inductive type. This produces in particular the ($n$-connected, $n$-truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions.
We propose foundations for a synthetic theory of $(infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary types. We define Segal types, in which binary composites exist uniquely up to homotopy; this automatically ensures composition is coherently associative and unital at all dimensions. We define Rezk types, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities - a local univalence condition. And we define covariant fibrations, which are type families varying functorially over a Segal type, and prove a dependent Yoneda lemma that can be viewed as a directed form of the usual elimination rule for identity types. We conclude by studying homotopically correct adjunctions between Segal types, and showing that for a functor between Rezk types to have an adjoint is a mere proposition. To make the bookkeeping in such proofs manageable, we use a three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which can be abstracted over using extension types that generalize the path-types of cubical type theory. In an appendix, we describe the motivating semantics in the Reedy model structure on bisimplicial sets, in which our Segal and Rezk types correspond to Segal spaces and complete Segal spaces.
We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and significant amounts of synthetic homotopy theory, as well as category theory and modalities. The library has been used as a basis for several independent developments. We discuss the decisions that led to the design of the library, and we comment on the interaction of homotopy type theory with recently introduced features of Coq, such as universe polymorphism and private inductive types.
We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of adjoint logic in which the discretization and codiscretization modalities are characterized using a judgmental formalism of crisp variables. This yields type theories that we call spatial and cohesive, in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the fundamental $infty$-groupoid or shape), disentangling the identifications of Homotopy Type Theory from the continuous paths of topology. In a further refinement called real-cohesion, the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwers fixed-point theorem.
We show that the classifying topos for the theory of fields does not satisfy De Morgans law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
