In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by several key results from the theory of quantaloid-enriched categories, this paper develops all needed ingredients purely in order-theoretic languages for the rea...

Current graphical object-oriented design notations are syntax-bound and semantic-free since they tend to focus on design representation rather than on the meaning of the design. This paper proposes a meaning for object-oriented designs in terms of object behaviours represented as constructions in category theory. A new design language is proposed, based on-notation, whose semantics is given by ...

It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case of ordered sheaves on a small quantaloid. Comparing module-equivalence with sheaf-equivalence for quantaloids and using the notion of centre of a quantaloid, ...

AM89] and JNW94] present abstract concepts of bisimulation in terms of category theory. This paper discusses the diierences between their formalisms and deals with the question how to translate these approaches into one another.

1 Moduli problems, spaces, and stacks. Vector bundles and K-theory 3 1.1 Some category theory . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Back to moduli spaces . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The way out of the problem . . . . . . . . . . . . . . . . . . . 7 1.4 Algebraic stacks and moduli of vector bundles . . . . . . . . . 7 1.5 K-theory of schemes . . . . . . . . . . ...

In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories.

Structure is important in large speci cations for understanding, testing and managing change. Category theory has been explored as framework for providing this structure, and has been successfully used to compose speci cations. This work has typically adopted a \correct by construction" approach: components are speci ed, proved correct and then composed together in such a way to preserve their ...

RM-ODP is about modeling, and mathematics is one of the oldest, and deserved, modeling disciplines. Experience accumulated in mathematics in general, and some of machineries developed, may well turn out to be relevant for arranging/organizing/ formalizing the concepts managed in RM-ODP. Especially promising here is mathematical category theory that is nothing but a discipline and framework for ...

The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to folklore but di cult to nd in the literature. While the results presented here are not new, our approach to ...