In [1] an equivalence of the categories SP and Cls was proven. The category SP consists of the state property systems [2] and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties [3, 4, 5, 6, 7, 8]. The category Cls consists of the closure spaces and the continuous maps. In earlier work it has been shown, using the equivalence between Cls and SP, that some of the axioms of quantum axiomatics are equivalent with separation axioms on the corresponding closure space. More particularly it was proven that the axiom of atomicity is equivalent to the T1 separation axiom [9]. In the present article we analyze the intimate relation that exists between classical and nonclassical in the state property systems and disconnected and connected in the corresponding closure space, elaborating results that appeared in [10, 11]. We introduce classical properties using the concept of super selection rule, i.e. two properties are separated by a superselection rule iff there do not exist ‘superposition states ’ related to these two properties. Then we show that the classical properties of a state property system correspond exactly to the clopen subsets of the corresponding closure space. Thus connected closure spaces correspond precisely to state property systems for which the

The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on spaces is an f-localization for some map f. We prove that both questions have an affirmative answer assuming the validity of a suitable large-cardinal axiom from set theory (Vopěnka’s principle). We also show that it is impossible to prove that all homotopy idempotent functors are f-localizations using the ordinary ZFC axioms of set theory (Zermelo–Fraenkel axioms with the axiom of choice), since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable, which is consistent with ZFC.

Monads are by now well-established as programming construct in functional languages. Recently, the notion of “Arrow ” was introduced by Hughes as an extension, not with one, but with two type parameters. At first, these Arrows may look somewhat arbitrary. Here we show that they are categorically fairly civilised, by showing that they correspond to monoids in suitable subcategories of bifunctors C op ×C → C. This shows that, at a suitable level of abstraction, arrows are like monads — which are monoids in categories of functors C → C. Freyd categories have been introduced by Power and Robinson to model computational effects, well before Hughes ’ Arrows appeared. It is often claimed (informally) that Arrows are simply Freyd categories. We shall make this claim precise by showing how monoids in categories of bifunctors exactly correspond to Freyd categories.

CatC of internal categories and functors in a given finitely complete categoryC. Several non-equivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C. Under mild conditions on C, the regular epimorphism topology determines a modelstructure where we is the class of weak equivalences of internal categories (in the sense of Bunge and Par'e). For a Grothendieck topos C we get a structure that, thoughdifferent from Joyal and Tierney's, has an equivalent homotopy category. In case C is semi-abelian, these weak equivalences turn out to be homology isomorphisms, and themodel structure on CatC induces a notion of homotopy of internal crossed modules. Incase C is the category

We introduce nonabelian differential cohomology classifying ∞-bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian n-group B n−1 U(1). Notable examples are String 2-bundles [9] and Fivebrane 6-bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spin-structures to String-structures [13] and further to Fivebrane-structures [133, 52], are abelian Chern-Simons 3- and 7-bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞-Lieintegrating the L∞-algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2- and twisted Fivebrane 6-bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted K-theory. We explain the Green-Schwarz mechanism in heterotic string theory in terms of twisted String 2-bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6-bundles. We close by transgressing differential cocycles to mapping

Abstract. Motivated by a desire to gain a better understanding of the “dimensionby-dimension” decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describe a suitable subcategory of Cat over which we can view the assignment C ↦ → Mnd(C) as an indexed category; on this base category, there is a natural topology. Then we single out a class of monads which are well-behaved with respect to reindexing. The main result is now, that such monads form a stack. Using this, we can shed some light on the free strict ω-category monad on globular sets and the free operad-with-contraction monad on the category of collections.

Abstract. In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and functor categories with a triple, in the last case we find examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications, we prove the existence of filtered minimal models for cdg algebras over a zero-characteristic field and we formulate an acyclic models theorem for non additive functors.

property systems and closure spaces:

This chapter is an introduction to the basic concepts, constructions, and results concerning locales. Locales (frames) are the object of study of the so called pointfree topology. They sufficiently resemble the lattices of open sets of topological spaces to allow the treatment of many topological questions. One motivation for the theory of locales is building topology on the intuition of “places of non-trivial extent ” rather than on points. Not the only one; hence it is not surprising that the theory has developed beyond the purely geometric scope. Still, we can think of a locale as of a kind of space, more general than the classical one, allowing us to see topological phenomena in a new perspective. Other aspects are, for instance, connections with domain theory [53, 52], continuous lattices [5, 31], logic [65, 20] and topos theory [42, 20]. Modern topology originates, in principle, from Hausdorff’s “Mengenlehre ” [30] in 1914. One year earlier there was a paper by Caratheodory [23] containing the idea of a point as an entity localized by a special system of diminishing sets; this is also of relevance for the modern point-free thinking. In the twenties and thirties the importance of (the lattice of) open sets (which are, typically, “places of non-trivial extent”) became gradually more and more apparent (see e.g. Alexandroff [1] or Sierpinski [54]). In [57] and [58], Stone presented his famous duality theorem from which it followed that compact zero-dimensional spaces and continuous maps are well represented by the Boolean algebras of closed open sets and lattice homomorphisms. This was certainly an encouragement for those who endeavoured to treat topology other than as a structure on a given system of points