Related papers: Introduction to Abstract Homotopy Theory

Interactions in complex systems are widely observed across various fields, drawing increased attention from researchers. In mathematics, efforts are made to develop various theories and methods for studying the interactions between spaces.…

The existence of a model structure on the category $\mathcal{D}$ of diffeological spaces is crucial to developing smooth homotopy theory. We construct a compactly generated model structure on the category $\mathcal{D}$ whose weak…

We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.

We study the homotopy type of the simplicial set of continuous semi-algebraic simplexes of an algebraic variety defined over a real closed field, which we will call the real homotopy type. We prove an analogue of the theorem of Artin-Mazur…

In previous work, the first author defined homotopy theories for stratified spaces from a simplicial and a topological perspective. In both frameworks stratified weak-equivalences are detected by suitable generalizations of homotopy links.…

Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The previous…

The recently introduced A-homotopy groups for graphs are investigated. The main concern of the present article is the construction of an infinite cell complex, the homotopy groups of which are isomorphic to the A-homotopy groups of the…

The purpose of this text is the study of the class of homotopy types which are modelized by strict \infty-groupoids. We show that the homotopy category of simply connected \infty-groupoids is equivalent to the derived category in…

Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some…

This paper is an expanded version of two talks given by the author at the Summer School on the Interactions between Homotopy Theory and Algebra at the University of Chicago, July 26 to August 6, 2004. It describes a connection between model…

Stable homotopy theory is governed by the principle that after inverting loop spaces, homotopy types become the representing objects for homology theories. We show that this principle extends to higher category theory: inverting…

We bring an abstract model theory perspective to interpolation. We ask, what is the role of interpolation in the study of extensions of first order logic, such as infinitary logics, generalized quantifiers and higher order logics? The…

In this paper we construct a cofibrantly generated model category structure on the category of all small symmetric multicategories enriched in simplicial sets.

We study the $2$-categories BIon, of (generalized) bounded ionads, and $\text{Acc}_\omega$, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex)…

This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension…

In the last year of his life, Bob Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure…

We construct combinatorial model category structures on the categories of (marked) categories and (marked) pre-additive categories, and we characterize (marked) additive categories as fibrant objects in a Bousfield localization of…

We introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe''…

Motivated by gauge theory, we develop a general framework for chain complex valued algebraic quantum field theories. Building upon our recent operadic approach to this subject, we show that the category of such theories carries a canonical…

It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental…