Flat topological modules and a new $\mathrm{Tor}$ functor

The subject of the talk falls into the same research area as my previous talk on flat modules (May 13, 2022). However, I will try to make today's talk reasonably self-contained and to repeat all necessary preliminaries. As before, our main objects will be flat locally convex modules, whose "correct" definition (as was shown at the previous talk) differs from the standard definition of a flat Banach (or Fréchet) module. This time we will discuss not merits, but a (quasi-)demerit of the new definition of flatness. It turns out that, in general, flat topological modules cannot be characterized in terms of the derived functor $\mathrm{Tor}$ (in contrast to flat modules in the purely algebraic setting, flat Banach modules, and flat Fréchet modules). We will discuss a possible way out of this situation which involves a new version of the $\mathrm{Tor}$ functor with values in the heart of a suitable t-structure on the derived category of complete locally convex spaces.