Michael Shulman

Univalence for inverse diagrams and homotopy canonicity

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodskys univalent model in simplicial sets, we obtain new models of univalence in a number of (∞, 1)-toposes; this answers a question raised at the Oberwolfach workshop on homotopical type theory. Second, by gluing the syntactic category of univalent type theory along its global sections functor to groupoids, we obtain a partial answer to Voevodskys homotopy-canonicity conjecture: in 1-truncated type theory with one univalent universe of sets, any closed term of natural number type is homotopic to a numeral.

Univalent Categories and the Rezk Completion

When formalizing category theory in traditional set-theoretic foundations, a significant discrepancy between the foundational notion of “sameness” – equality – and its practical use arises: most category-theoretic concepts are invariant under weaker notions of sameness than equality, namely isomorphism in a category or equivalence of categories. We show that this discrepancy can be avoided when formalizing category theory in Univalent Foundations.

Calculating the Fundamental Group of the Circle in Homotopy Type Theory

Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lofs dependent type theory and the mathematical disciplines of category theory and homotopy theory. The mathematics suggests new principles to add to type theory, while the type theory can be used in novel ways to do computer-checked proofs in a proof assistant. In this paper, we formalize a basic result in algebraic topology, that the fundamental group of the circle is the integers. Our proof illustrates the new features of homotopy type theory, such as higher inductive types and Voevodskys univalence axiom. It also introduces a new method for calculating the path space of a type, which has proved useful in many other examples.

Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative” traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow”. In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate “cylindrical” type of string diagram, which we justify formally in an appendix.

Lectures on N-Categories and Cohomology

This is an explanation of how cohomology is seen through the lens of n-category theory. Special topics include nonabelian cohomology, Postnikov towers, the theory of ‘n-stuff’, and n-categories for n = −1 and −2. A lengthy appendix clarifies certain puzzles and ventures into deeper waters such as higher topos theory. An annotated bibliography provides directions for further study.

Enhanced 2-categories and limits for lax morphisms

Abstract We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is “enhanced”, by specifying which of these weak morphisms are actually strict, can be thought of as category enriched over a particular base cartesian closed category F . We give a complete characterization, in terms of F -enriched category theory, of the limits which exist in such 2-categories of categories with extra structure.

The HoTT library: a formalization of homotopy type theory in Coq

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.

Quantum Gauge Field Theory in Cohesive Homotopy Type Theory.

We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by the authors developed in detail elsewhere.

The multiplicativity of fixed point invariants

We prove two general factorization theorems for fixed-point invariants of fibrations: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar multiplicativity results for the Lefschetz and Nielsen numbers of a fibration. Moreover, the proofs of these theorems are essentially formal, taking place in the abstract context of bicategorical traces. This makes generalizations to other contexts straightforward. 55M20; 18D05, 55R05

Parametrized spaces model locally constant homotopy sheaves

Abstract We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopy-theoretic version of the correspondence between covering spaces and π 1 -sets. We then use these two equivalences to study base change functors for parametrized spaces.