Received We show how to solve the word problem for simply typed-calculus by using a few well-known facts about categories of presheaves and the Yoneda embedding. The formal setting for these results is P-category theory, a version of ordinary category theory where each hom-set is equipped with a partial equivalence relation. The part of P-category theory we develop here is constructive and thus...

This paper proposes a programme of research with the aim of understanding network-based systems through the application of monoidal category theory, and outlines some in-progress work in this direction.

This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them with extra structure such as a tensor product and a triangulated category structure. We discuss the construction of the Baum– Connes assembly map via localisa...

We propose for the Effective Topos an alternative construction: a realisability framework composed of two levels of abstraction. This construction simplifies the proof that the Effective Topos is a topos (equipped with natural numbers), which is the main issue that this paper addresses. In this our work can be compared to Frey’s monadic tripos-to-topos construction. However, no topos theory or ...

Along the lines of classical categorical type theory for total functions, we establish correspondence results between certain classes of partial equational theories on the one hand and suitable classes of categories having certain finite limits on the other hand. E.g., we show that finitary partial theories with existentially conditioned equations are essentially the same as cartesian categorie...

We examine the need for a new approach to science that takes into account both structural science, i.e. current science as practiced by most scientists, and structural-phenomenological science, which allows for the coupling of quan-tum theory to brain functions, examines the nature of consciousness, etc. This new approach will result in what is termed integrative science and, we postulate, is t...

On the surface, the emergent descriptive theory of represented spaces is the extension of both classical [36] and effective [52] descriptive set theory from its usual setting of Polish spaces to the much larger class of represented spaces. There is far more to it, though: By extending the setting, suddenly connections to areas such as category theory (with a certain topos-theoretic flavour) and...

This paper presents a novel technique of using Prolog with never instantiated variables to manipulate a range of algebraic structures. The paper argues that Prolog is a powerful and underrated tool for use in computational number theory. A detailed example is presented in this extended abstract, and several in the full paper, showing the advantages of using this technique. The detailed example ...

The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic speciications. In detail we discuss diierent approaches to an abstract theory of spec-iication logics. Further we present a uniform framework for developing particular speciication logics. We make use of`classifying categories', to present categories of algebr...

lthough fuzzy set theory and  sheaf theory have been developed and studied independently,  ulrich hohle shows that a large part of fuzzy set  theory  is in fact a subfield of sheaf theory. many authors have studied mathematical structures, in particular, algebraic structures, in both  categories of these generalized (multi)sets. using hohle's idea, we show that for a (universal) algebra $a$, th...