2 Categories and examples 7 2.1 The definition of a category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Categories as graphs with composition . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Categories as partial semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Categories as enriched categories . . . . . . . . . . . . . . . . . . . . . . . . ....

In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categori...

We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally-enriched functor from a spectrally-enriched version of permutative categories to the category of spectra that is ...

This paper presents an investigation of many valued lattices from the point of view of enriched category theory. For a bounded partially ordered set P , the conditions for P to become a lattice can be postulated as existence of certain adjunctions. Reformulating these adjunctions, by aid of enriched category theory, in many valued setting, two kinds of many valued lattices, weak -lattices and -...

In this paper, projective modules over a quantale are characterized by distributivity, continuity, and adjointness conditions. It is then show that a morphism Q // A of commutative quantales is coexponentiable if and only if the corresponding Q-module is projective, and hence, satisfies these equivalent conditions.

Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category V its associat...

1.3. De nition. ABSTRACT. We apply enriched category theory to study Cauchy completeness in continuity spaces. Our main result is the equivalence in continuity spaces of the category theoretic and the uniform notions of Cauchy completeness. This theorem, which generalizes a result of Lawvere for quasi-metric spaces, makes a natural connection between the category-theoretic and topological aspec...

The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares....

Our work is a fundamental study of the notion of approximation inQ-categories and in (U,Q)-categories. Our exposition is categorical but kept as close to language of domain theory as possible. Consequently, we introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize J-continuous Q-categories (...

We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where all objects of V are cofibrant.