We construct natural equivalences between derived categories of coherent sheaves on the local models for stratified Mukai or Atiyah flops (of type A).

Image formation is described by means of modern geometry in terms of topological and geometric curvatures. These curvatures and derived equivalences are a quantisation of the set of rules for constructing the image in particular along discontinuity, singularity and bifurcation sets. A simpliication of the set of construction rules is proposed by applying a dynamic scale-space theory. Dynamic he...

A given monoid usually admits many presentations by generators and relations the notion of Tietze equivalence characterizes when two describe same monoid: it is case one can transform presentation into other using families so-called transformations. The goal this article to provide an abstract geometrical understanding well-known fact, constructing a model structure on category presentations, i...

The question that serves as the title is to be understood in the context of Quillen model categories: having fixed the category of models and the subcategory of weak equivalences, how much ambiguity is there in finding suitable fibration and cofibration classes? Already for simplicial sets, infinitely many different subclasses of the monomorphisms can serve as cofibrations (with the usual weak ...