Related papers: On the subdivision of small categories

In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model…

In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian…

This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…

We show that the categories of compact Lie groups and complex reductive groups (not necessarily connected) are homotopy equivalent topological categories. In other words, the corresponding categories enriched in the homotopy category of…

We construct a category that classifies compact Hausdorff spaces by their shape and finite topological spaces by their weak homotopy type.

We prove a result that enables us to calculate the rational homotopy of a wide class of spaces by the theory of minimal models.

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is…

Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…

The concept of relative sectional category expands upon classical sectional category theory by incorporating the pullback of a fibration along a map. Our paper aims not only to explore this extension but also to thoroughly investigate its…

Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…

A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. As a homotopy invariant, the homotopy set of…

In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of $X$ is the universal covering in categorical sense.…

We algorithmically compute integral Eilenberg-MacLane homology of all semigroups of order at most $8$ and present some particular semigroups with notable classifying spaces, refuting conjectures of Nico. Along the way, we give an…

By introducing various topologies on the homotopy groups of a topological space, some researchers make these well known notions in algebraic topology more useful and powerful. In this paper, first we recall and review some known topologies…

Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as…

We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories.

Let G be a finite group and p a prime dividing its order. We define new collections of p-subgroups of G. We study the homotopy relations among them and with the standard collections of p-subgroups. We determine their ampleness and sharpness…

Extriangulated categories axiomatize extension-closed subcategories of triangulated categories. We show that the homotopy category of an exact quasi-category can be equipped with a natural extriangulated structure.

We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…