Weak fibration categories - theory and applications.
Model categories, introduced by Quillen, provide a very general context in which it is possible to set up the basic machinery of homotopy theory. In particalar they enable to define derived functors, homotopy limits and colimits, cohomology theories and spectral sequences to catculate them. However, the structure of a model category is usually hard to verify, and in some interesating cases even impossible to define. In this lecture I will define a much simpler notion then a model category, called a weak fibration category. By a theorem due to T. Schlank and myself, a weak fibration category gives rise in a natural way to a model category structure on its pro category, provided some technical assumptions are satisfied. This result can be used to construct new model structures in different mathematical fields, and thus to import the methods of homotopy theory to these situations. Examples will be given from the categories of simplicial presheaves, C*-algebras and complexes in Abelian categories. Applications will be discussed with each example.
The above encompasses joint work with Tomer M. Schlank, Yonatan Harpaz, Geoffroy Horel, Michael Joachim Snigdhayan Mahanta and Matan Prezma.