Bimorphisms in pro-homotopy and proper homotopy

A Simplicial Model for Proper Homotopy Types

The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of controlled sets, yielding simplicial controlled sets, such that one can functorially produce a singular simplicial controlled set CSing(MaxCtl(X)) from a locally compact X. We then argue that this CSing(MaxCtl(X)) captures the (weak) proper homotopy type of X. Moreover, our te...

Graph homotopy and Graham homotopy

Simple-homotopy for simplicial and CW complexes is a special kind of topological homotopy constructed by elementary collapses and expansions. In this paper we introduce graph homotopy for graphs and Graham homotopy for hypergraphs, and study the relation between these homotopies and the simplehomotopy for simplicial complexes. The graph homotopy is useful to describe topological properties of d...

Equivariant Homotopy Epimorphisms, Homotopy Monomorphisms and Homotopy Equivalences

Homology and Homotopy Theory Homotopy and Homology

1. TJ'he first step toward connecting these two basic concepts of topology was taken by L. E. J. Brouwer in 1912 by demonstrating that two continuous mappings of a two-dimensional sphere into itself can be continuously deformed into each other if and only if they have the same degree (that is, if they are equivalent from the point of view of homology theory). After having generalized Brouwer's ...

Homotopy Theory of Homotopy Algebras

The goal of this paper is to study the homotopy theory of homotopy algebras over a Koszul operad with their infinity morphisms. The method consists in endowing the category of coalgebras over the Koszul dual cooperad with a model category structure. CONTENTS Introduction 1 1. Recollections 2 2. Model category structure for coalgebras 5 3. Conclusion 14 Appendix A. A technical lemma 15 Reference...