The derived categories of the LB-spaces

Let LB be the category of LB-spaces, which has as objects precisely those Hausdorff locally convex spaces that can be written as a countable inductive limit of Banach spaces, and as morphisms the continuous linear maps between them. In the talk we will firstly review LB’s categorical properties and explain its place in the general hierarchy of non-abelian categories. After that we will show that there are two natural ways to define a derived category of LB which however lead to non-equivalent homological theories.