We discuss two aspects of the presentation of the theory of principal ∞-bundles in an ∞-topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal ∞-bundles over any object in the ∞-topos over C are classified by hyper-Čech-cohomology with coefficients in G. T...

An abstract deenition of bisimulation is presented. It enables a uniform deenition of bisimulation across a range of diierent models for parallel computation presented as categories. As examples, transition systems, synchronisation trees, transition systems with independence (an abstraction from Petri nets) and labelled event structures are considered. On transition systems the abstract deeniti...

Definition 1. A T -prespectrum E is a sequence En of pointed spaces together with structure morphisms σn : En ∧ T → En+1 for all n. A morphism λ : E → F of T -prespectra is a sequence of maps λn : En → Fn commuting with the structure morphisms. We will denote the category of T -prespectra by PSpT (k). Example 1. For any based space X , we have the suspension prespectrum Σ∞(X ) of X given by Σ∞(...

The main theorem of [4] say that there is a proper closed simplicial model category structure on the category Spt of symmetric spectra, and that the associated homotopy category Ho(Spt) is equivalent to the stable category. This paper shows how to generalize this result to the category PreSpt(C) of presheaves of symmetric spectra on an arbitrary small Grothendieck site C. The reader will observ...

We use dialgebras, generalising both algebras and coalgebras, as a complement of the standard coalgebraic framework, aimed at describing the semantics of an interactive system by the means of reaction rules. In this model, interaction is built-in, and semantic equivalence arises from it, instead of being determined by a (possibly difficult) understanding of the side effects of a component in is...

Let k be an algebraically closed field of characteristic zero. Let c : SH → SH(k) be the functor induced by sending a space to the constant presheaf of spaces on Sm/k. We show that c is fully faithful. In particular, c induces an isomorphism c∗ : πn(E)→ Πn,0(c(E)) for all spectra E. Fix an embedding σ : k → C and let ReB : SH(k)→ SH be the associated Betti realization. Let Sk be the motivic sph...

A generalization of the usual ideles group is proposed, namely, we construct certain adelic complexes for sheaves of K-groups on schemes. More generally, such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated to the presheaf of a homology theory with certain natural axioms, satisfied by K-theory. In this case it is proven that the adelic ...

These are some lecture notes for a course presenting the cubical set model of type theory, first in Copenhagen, December 2014, and then in Paris, February 2015. We describe a particular presheaf model of type theory. This description can also be seen as an operational semantics of a purely syntactical type system. It involves a nominal extension of λ-calculus. We use a generalization of the Kan...

The internal structure of a measuring device, which depends on what its components are and how they are organized, determines how it categorizes its inputs. This paper presents a geometric approach to studying the internal structure of measurements performed by distributed systems such as probabilistic cellular automata. It constructs the quale, a family of sections of a suitably defined preshe...