Abstract We describe a new sequence of polytopes which characterize A∞-maps from a topological monoid to an A∞-space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the n...

This talk reports on ongoing work – with Tony Hoare, Akbar Hussain, Bernhard Möller, Rasmus Petersen, Georg Struth, Ian Wehrman, and others – on models and logics for concurrent processes [10,6,5]. The approach we are taking abstracts from syntax or particular models. Message passing and shared memory process interaction, and strong (interleaving) and weak (partial order) approaches to sequenci...

In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these can be constructed from the groupoid quotient. define an internal notion of signatures for HITs, and each signature, construct a bicategory algebras 1-types groupoids. continue proving initial algebra semantics our signatures. After that, show quotient induces b...

We introduce a data-structure problem on graphs we call the bicate-gory problem, and a data structure that solves it in O(p m logm) amor-tized time, where m is the number of edges. We show how this data structure can be used in quickly computing approximately minimum-cost networks. The resulting time bound for the approximation algorithm is O(n p m logm) where n and m are the number of nodes an...

This paper studies fundamental connections between profunctors (i.e., distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view ...

In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel’d double. If A is a map pseudomonoid in a monoidal bicategory M , the analogue of the category of Hopf modules for A is an Eilenberg-Moore construction for a certain monad in Hom(M op,Cat). We study the existence of the ...

We construct a category of circuits: the objects are alphabets and the morphisms are deterministic automata. The construction diiers in several respects from the bicategories of circuits appearing previously in the literature: it is parameterized by a monad which allows exibility in the emergent notion of process. We focus on the circuits which arise from a distributive category and the excepti...

There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened — these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category ∆ is replaced by a certain ‘fat’ delta of ‘coloured ordinals’, where the d...

Kornel Szlachányi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudo-monoids) in any monoidal bicategory M . These are...

In this paper we answer the question: ‘what kind of a structure can a general multicategory be enriched in?’ (Here ‘general multicategory’ is used in the sense of [Lei1], [Bur] or [Her].) The answer is, in a sense to be made precise, that a multicategory of one type can be enriched in a multicategory of the type one level up. In particular, we will be able to speak of a Tn-multicategory enriche...