We show that unstable D-branes play the role of “D-sphalerons” in string theory. Their existence implies that the configuration space of type-II string theory has a complicated homotopy structure, similar to that of an infinite grassmannian. In particular, the configuration space of type-IIA (-IIB) string theory on R has non-trivial homotopy groups πk for all k even (odd).

This thesis is composed of three separate parts. The first part deals with definability and productivity issues of equational systems defining polymorphic stream functions. The main result consists of showing such systems composed of only unary stream functions complete with respect to specifying computable unary polymorphic stream functions. The second part deals with syntactic and semantic no...

We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment includes Voevodsky’s univalence axiom and a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cu...

Following the cubical set model of type theory which validates the univalence axiom, cubical type theories have been developed that interpret the identity type using an interval pretype. These theories start from a geometric view of equality. A proof of equality is encoded as a term in a context extended by the interval pretype. Our goal is to develop a cubical theory where the identity type is...

Homotopy type theory is a new foundation of mathematics under current development. To compare it with the existing set theoretic foundation, we formalize the cumulative hierarchy of sets in the Coq system, closely following and clarifying the informal treatment in the homotopy type theory book [17].

We prove that any stable, additive, combinatorial model category M has a canonical model enrichment over Sp(sAb) (symmetric spectra based on simplicial abelian groups). So to any object X ∈ M one can attach an endomorphism ring object in this category, denoted hEndad(X). Since the homotopy theory of ring objects in Sp(sAb) is equivalent to the homotopy theory of differential graded algebras, on...

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a ‘system of open neighborhoods at infinity’ while an exterior map is a continuous map which is ‘continuous at infinity’. The category of spaces and proper maps is a subcategory of the category of exterior spaces. In this paper we show that the category of e...

In this paper we are interested in finite connected 2-dimensional CTF-complexes, each with a single 2-celL We show any two such complexes have the same homotopy type if their fundamental groups are isomorphic. In fact, there is a homotopy equivalence inducing any isomorphism of the fundamental groups. We also study the homotopy factorizations of such spaces into finite sums. In this paper we ar...

Martin-Löf’s intuitionistic type theory is a widely-used framework for constructive mathematics and computer programming. In its most popular form, type theory consists of a collection of inference rules inductively defining formal proofs. These rules are justified by Martin-Löf’s meaning explanations, which extend the Brouwer-Heyting-Kolmogorov interpretation of connectives to a rich collectio...

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbif...