The main goal of this thesis is to define an extension of Gödel not-not translation to all truncated types, in the setting of homotopy type theory. This goal will use some existing theories, like Lawvere-Tierney sheaves theory in toposes, we will adapt in the setting of homotopy type theory. In particular, we will define a Lawvere-Tierney sheafification functor, which is the main theorem presen...

Morphoid type theory (MTT) is a typetheoretic foundation for mathematics supporting the concept of isomorphism and the substitution of isomorphics. Unlike homotopy type theory (HoTT), which also supports isomorphism, morphoid type theory is a direct extension of classical predicate calculus and avoids the intuitionistic constructs of propositionsas-types, path induction and squashing. Although ...

Wepresent a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute in many cases. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built...

In the classical paper [A-H] Adams-Hilton constructed a free chain algebra which is an important algebraic model of a simply connected homotopy type. We show that this chain algebra (endowed with an additional structure given by a “height function”) yields actually an invariant of a proper homotopy type. For this we introduce the homotopy category of locally finite chain algebras without using ...

In this paper we consider lightweight routing in a wireless sensor network deployed in a complex geometric domain Σ with holes. Our goal is to find short paths of different homotopy types, i.e., paths that go around holes in different ways. In the example of Figure 1, there are three holes in the network and there are many different ways to “thread” a route from s to t. Observe that paths α, β,...

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened “homotopy type theory”. In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, cal...

We introduce a notion of ‘cover of level n’ for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram of n-covers biject with global homotopy classes when the target is an n-type. When the target is an Eilenberg–MacLane sheaf, this specializes to computing derived functor cohomology, up to degree ...

A simply connected topological space X has homotopy Lie algebra ( X) Q. Following Quillen, there is a connected di erential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X , and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property we call separated. The homo...

1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4 (iv))) as a general framework for “doing homotopy theory”. To each model category M there is associated a homotopy category. If W C M denotes the subcategory of the weak equivalences, then this homot...

We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invaria...