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...

We thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloidQ. In analogy with V-category theory we discuss such things as adjoint functors, (pointwise) left Kan extensions, weighted (co)limits, presheaves and free (co)completion, Cauchy completion and Morita equivalence. With an appendix on the uni...

In this first of two papers on ∞-category theory, we present information and resources on the basic prerequisites to parse the Riehl-Verity theory of ∞-cosmoi, the main topic of, e.g, [RV16b]. We provide basic definitions of monoidal categories, enriched categories, model categories, simplicial sets, and quasicategories, leading up to the main definition of [RV17], that of an ∞cosmos. Examples ...

Abstract We introduce a general definition for coloured cyclic operads over symmetric monoidal ground category, which has several appealing features. The forgetful functor from to both adjoints, each of is relatively simple. Explicit formulae these adjoints allow us lift the Cisinski–Moerdijk model structure on category enriched in simplicial sets sets.

We prove that for each locally $\alpha$-presentable category $\mathcal K$ there exists a regular cardinal $\gamma$ such any $\alpha$-accessible functor out of (into another category) is continuous if and only it preserves $\gamma$-small limits; as consequence we obtain new adjoint theorem specific to the functors K$. Afterwards generalize these results enriched setting deduce, among other thing...

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory...

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if categorical structure question is sufficiently weak (such as monoidal, but not strict categories) then 2-category accessible. Furthermore, explore flexible limits such possess interaction with filtered colimits.

In the setting of enriched category theory, we describe dual adjunctions of the form L R : Spa −→ Alg between the dual of the category Spa of “spaces” and the category Alg of “algebras” that arise from a schizophrenic object , which is both an “algebra” and a “space”. We call such adjunctions logical connections. We prove that the exact nature of is that of a module that allows to lift optimall...

Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadic...

Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadic...