In a paper of 1974, Brian Day employed a notion of factorization system in the context of enriched category theory, replacing the usual diagonal lifting property with a corresponding criterion phrased in terms of hom-objects. We set forth the basic theory of such enriched factorization systems. In particular, we establish stability properties for enriched prefactorization systems, we examine th...

We revisit sheaves on locales by placing them in the context of the theory of Hilbert quantale modules. The local homeomorphisms p : X → B are identified with the locales X that are Hilbert B-modules equipped with a natural notion of basis. These modules form a full subcategory B-HMB of the category of Hilbert B-modules where all the homomorphisms are adjointable: for each homomorphism h there ...

Let R be a (unital) commutative ring, andG be a finite group with order invertible in R. We introduce new idempotents εT,S in the double Burnside algebra RB(G,G) of G over R, indexed by conjugacy classes of minimal sections (T, S) of G (i.e. sections such that S ≤ Φ(T )). These idempotents are orthogonal, and their sum is equal to the identity. It follows that for any biset functor F over R, th...

We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith’s result to le...

In [6] Higson showed that the formal properties of the Kasparov KK -theory groups are best understood if one regards KK (A, B) for separable C∗-algebras A, B as the morphism set of a category KK . In category language the composition and exterior KK product give KK the structure of a symmetric monoidal category which is enriched over abelian groups. We show that the enrichment of KK can be lift...

We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. due to secrecy and deceit, as well as nested knowledge. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the c...

We develop a formal system to reason about knowledge properties of quantum security protocols. The formalism is obtained via a marriage of measurement calculus [3], an algebraic framework for measurementbased quantum computing [7], with the algebra of epistemic actions and their appearance maps [1, 8]. Measurement calculus has also proven to be a proper language to describe and to analyse distr...

In this paper, we study the relations (L, )-fuzzy topologies and (L, )-filters on strictly two-sided, commutative quantale lattices (L, ) and (L, ∗). Mathematics Subject Classification: 03E72, 54A40, 54B10

In this paper, we study (L, )-fuzzy topologies induced by operations and (L, )-filters on strictly two-sided, commutative quantale lattices (L, ) and (L, ∗). Mathematics Subject Classification: 03E72, 54A40, 54B10

We present two applications to AI of recently introduced high level quantum structures. These structures are the categorical quantum logic of (Abramsky & Coecke 2004) and the quantale quantum logic of (Coecke, Moore, & Stubbe 2001). Firstly, we show how the diagrammatic toolkit of categorical quantum logic, when restricted to its pregroup fragment (Lambek 1999; 2001), simplifies analysis of sen...