Moscow Mathematical Journal

Volume 24, Issue 2, April–June 2024 pp. 219–286.

Exact Categories of Topological Vector Spaces with Linear Topology

Authors: Leonid Positselski (1)
Author institution:(1) Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1 (Czech Republic);
Laboratory of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow 127051 (Russia)

Summary:

We explain why the naïve definition of a natural exact category structure on complete, separated topological vector spaces with linear topology fails. In particular, contrary to Beilinson’s paper “Remarks on topological algebras” (Moscow Mathematical Journal 8:1 (2008), 1–20), 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.

2020 Math. Subj. Class. 22A05, 18E05, 18A30, 46A32.

Keywords: Exact categories, quasi-abelian categories, semiabelian categories, topological abelian groups, topological vector spaces, linear topology, incompleteness of quotients, maximal exact structure, pro-vector spaces, topological tensor products.

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