We combine two research directions of the past decade, namely the development of a lax-algebraic framework for categories of interest to topologists and analysts, and the exploration of key topological concepts, like separation and compactness, in an abstract category which comes equipped with an axiomatic notion “closed” or “proper” map. Hence, we present various candidates for such notions in...

Quasi-categories live at the intersection of homotopy theory with category theory. In particular, they serve as a model for (∞, 1)-categories, that is, weak higher categories with n-cells for each natural number n that are invertible when n > 1. Alternatively, an (∞, 1)-category is a category enriched in ∞-groupoids, e.g., a topological space with points as 0-cells, paths as 1-cells, homotopies...

There is an ongoing massive effort by many researchers to link category theory and geometry, especially homotopy coherence and categorical coherence. This constitutes just a part of the broad undertaking known as categorification as described by Baez and Dolan. This effort has as a partial goal that of understanding the categories and functors that correspond to loop spaces and their associated...

We continue work of our earlier paper [20] where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theore...

Abstract We construct and study projective Reedy model category structures for bimodules infinitesimal over topological operads. Both produce the same homotopy categories. For categories in question, we build explicit cofibrant fibrant replacements. show that these are right proper under some conditions left proper. also extension/restriction adjunctions.

A classification of open equivariant topological conformal field theories in terms Calabi-Yau $A_\infty $-categories endowed with a group action is presented.

inferences as metaphorical spatial inferences Spatial inferences are characterized by the topological structure of image-schemas. We have seen cases such as CATEGORIES ARE CONTAINERS and LINEAR SCALES ARE PATHS where image-schema structure is preserved by metaphor and where abstract inferences about categories and linear scales are metaphorical versions of spatial inferences about containers an...

The homotopy hypothesis was originally stated by Grothendieck [13] : topological spaces should be “equivalent” to (weak) ∞-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g. for rewriting, distributed systems, (homotopy) type theory etc. But an essential feature in the work set up in concurrency ...

The paper is about the comparison between (apparently) different cartesian closed extensions of the category of topological spaces. Since topological spaces do not in general allow formation of function spaces, the problem of determining suitable categories with such a property—and nicely related to that of topological spaces—was studied from many different perspectives: general topology, funct...

In this paper we investigate effective descent morphisms in categories of reflexive and transitive lax algebras. We show in particular that open and proper maps are effective descent, result that extends the corresponding results for the category of topological spaces and continuous maps. Introduction A morphism p : E → B in a category C with pullbacks is called effective descent if it allows a...