Exact categories of topological vector spaces with linear topology

Abstract

We explain why the naive definition of a natural exact category structure on complete, separated topological vector spaces with linear topology fails. In particular, contrary to arXiv:0711.2527, the category of such topological vector spaces is not quasi-abelian. We present a corrected definition of exact category structure which works OK. Then we explain that the corrected definition still has a shortcoming in that a natural tensor product functor is not exact in it, and discuss ways to refine the exact category structure so as to make the tensor product functors exact.

LaTeX 2e with xy-pic; 71 pages, 9 commutative diagrams; v.2: Remark 12.2, Example 13.1(3), and Remark 13.6 inserted; v.3: seven references added; v.4: several misprints corrected, references added and updated, two paragraphs inserted in the Introduction, one in Example 4.1, and one in Section 9; v.5: several misprints corrected