Homotopy epimorphisms in homotopy pullbacks

Equivariant Homotopy Epimorphisms, Homotopy Monomorphisms and Homotopy Equivalences

QUILLEN THEOREMS Bn FOR HOMOTOPY PULLBACKS OF (∞, k)-CATEGORIES

We extend the Quillen Theorem Bn for homotopy fibers of Dwyer et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative categories, not only with respect to their Reedy structures but also their Rezk structure, which turns them into models for the theories of (∞, 1)and (∞, k)-categories ...

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...

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...

Essential laminations and deformations of homotopy equivalences, II: the structure of pullbacks

It is a long-standing conjecture in 3-manifold theory that every homotopy equivalence f :M!N between closed, irreducible 3-manifolds M and N , with j 1(M)j (=j 1(N)j) = 1, is homotopic to a homeomorphism. In [Wa], Waldhausen proved that this conjecture was true, if we assume that N contains a 2-sided incompressible surface S. The proof consists of rst homotoping the map f so that the pullback s...