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Mathematics > Category Theory

Title: 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.
Comments: 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
Subjects: Category Theory (math.CT); General Topology (math.GN)
Journal reference: Moscow Math. Journal 24 #2 (2024), p.219-286
DOI: 10.17323/1609-4514-2024-24-2-219-286
Cite as: arXiv:2012.15431 [math.CT]
  (or arXiv:2012.15431v5 [math.CT] for this version)

Submission history

From: Leonid Positselski [view email]
[v1] Thu, 31 Dec 2020 03:52:40 GMT (59kb)
[v2] Mon, 4 Jan 2021 23:03:54 GMT (60kb)
[v3] Tue, 9 Mar 2021 20:39:28 GMT (61kb)
[v4] Tue, 26 Sep 2023 09:43:12 GMT (62kb)
[v5] Thu, 9 May 2024 15:13:50 GMT (62kb)

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