Topic на борде конференции
Feb. 10th, 2021 09:46 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
posichttps://cats2021.discussion.community/post/leonid-positselski-exact-categories-of-topological-vector-spaces-with-linear-topology-11383757?pid=1322567718
Leonid Positselski (Institute of Mathematics, Czech Academy of Sciences, Prague)
Exact categories of topological vector spaces with linear topology
If the rules of this forum allow that, I just wanted to post the link to my preprint here:
https://arxiv.org/abs/2012.15431
***
Thank you for the additional advertisement! Let me add that, besides the main topic of topological algebra with linear topology, my paper also contains an auxiliary discussion of additive categories, including a simple algebraic counterexample to Raikov's conjecture in Example 5.5. (Actually, it is taken from my previous paper [22].) Since you raised the question in another topic, let me add that Example 5.5 is an integral (but not quasi-abelian) category, since it has a faithful additive functor to an abelian category which preserves kernels and cokernels.
***
I was totally unaware of the existence of Raikov's conjecture or any work by other people related to Raikov's conjecture back in 2011. In fact, my paper published in Moscow Math. J. in 2011 was largely based on material which I originally wanted to include into my Ph.D. thesis (defended in 1998) as its main part, but was unable to either solve what I perceived as the main problem (about mixed Artin-Tate motives with finite coefficients), or even write up properly what I already knew about it.
In my approach to mixed Artin-Tate motives with finite coefficients (which are a concept from arithmetic geometry or something like that), such motives were supposed to form exact categories. That was one of the main differences from the rational coefficients case, where such motives are expected to form abelian categories.
I started to work on this problem in 1995, and the first thing I did was spending several months teaching myself the basics of exact categories, mostly not by looking into any literature but by redeveloping the theory for myself from scratch. So I rediscovered some results of Keller, in particular (such as the fact that Quillen's "obscure axiom" follows from the other axioms, or that it suffices to require, say, the admissible epimorphisms to be closed under the composition and the same property for the admissible monomorphisms then follows, etc.).
I published almost nothing at all for nine years between 1996-04, and then again for four years between 2006-09. Since 2006 I started to write up and publish my old ideas, and in particular in 2010 I returned to the topic of mixed Artin-Tate motives with finite coefficients and solved the main problem. Then I browsed through my archives, found all kinds of notes, going back to 1995-98, tangentially related to the would-be main part of the dissertation. Then I reworked and included most of them in the paper in MMJ-2011 (as appendices etc.).
I have an unfinished latex file in my archives, on my computer, which I typed during my graduate study years. It is dated September 1998 in the operating system. The file contains a paragraph which reads as follows:
"Example 1. The following counterexample shows that the condition Ex2''(c) is indeed necessary. Let $\A$ be the additive category whose objects are morphisms of vector spaces $f\:V''\rarrow V'$ endowed with a subspace $V\sub\Im f$ and let $\T_\A$ be the class of all triples for which Ex1 holds. One can check that this class of admissible triples satisfies Ex0, Ex2''(a-b) and Ex3, but not Ex2''(c)."
That was the context in which my counterexample to what later became known as Raikov's conjecture was invented back in the '90s. In 2010 I took this example from the old file and put it into the new paper, which was subsequently accepted by MMJ.
So I studied both what (as I later learned) people call quasi-abelian and semi-abelian categories, and also maximal exact category structures on weakly idempotent-complete additive categories, back in the '90s, independently of any work by other people on these topics. And I did not publish my findings until the 2011 paper (which appeared as an arXiv preprint in 2010).
My motivation was to understand exact categories well enough to prepare myself to study the exact categories of mixed Artin-Tate motives (and motivic sheaves) with finite coefficients. These form pretty complicated exact categories, and I thought I needed to develop intuition on simpler examples and counterexamples first.
Чувствую себя каким-то реликтом века назад ушедшей в прошлое эпохи. Когда можно было так работать. Сначала 15 лет думать, потом публиковать так, чтобы никто не заметил, потом еще 10 лет думать, потом начинать людям рассказывать.
Leonid Positselski (Institute of Mathematics, Czech Academy of Sciences, Prague)
Exact categories of topological vector spaces with linear topology
If the rules of this forum allow that, I just wanted to post the link to my preprint here:
https://arxiv.org/abs/2012.15431
***
Thank you for the additional advertisement! Let me add that, besides the main topic of topological algebra with linear topology, my paper also contains an auxiliary discussion of additive categories, including a simple algebraic counterexample to Raikov's conjecture in Example 5.5. (Actually, it is taken from my previous paper [22].) Since you raised the question in another topic, let me add that Example 5.5 is an integral (but not quasi-abelian) category, since it has a faithful additive functor to an abelian category which preserves kernels and cokernels.
***
I was totally unaware of the existence of Raikov's conjecture or any work by other people related to Raikov's conjecture back in 2011. In fact, my paper published in Moscow Math. J. in 2011 was largely based on material which I originally wanted to include into my Ph.D. thesis (defended in 1998) as its main part, but was unable to either solve what I perceived as the main problem (about mixed Artin-Tate motives with finite coefficients), or even write up properly what I already knew about it.
In my approach to mixed Artin-Tate motives with finite coefficients (which are a concept from arithmetic geometry or something like that), such motives were supposed to form exact categories. That was one of the main differences from the rational coefficients case, where such motives are expected to form abelian categories.
I started to work on this problem in 1995, and the first thing I did was spending several months teaching myself the basics of exact categories, mostly not by looking into any literature but by redeveloping the theory for myself from scratch. So I rediscovered some results of Keller, in particular (such as the fact that Quillen's "obscure axiom" follows from the other axioms, or that it suffices to require, say, the admissible epimorphisms to be closed under the composition and the same property for the admissible monomorphisms then follows, etc.).
I published almost nothing at all for nine years between 1996-04, and then again for four years between 2006-09. Since 2006 I started to write up and publish my old ideas, and in particular in 2010 I returned to the topic of mixed Artin-Tate motives with finite coefficients and solved the main problem. Then I browsed through my archives, found all kinds of notes, going back to 1995-98, tangentially related to the would-be main part of the dissertation. Then I reworked and included most of them in the paper in MMJ-2011 (as appendices etc.).
I have an unfinished latex file in my archives, on my computer, which I typed during my graduate study years. It is dated September 1998 in the operating system. The file contains a paragraph which reads as follows:
"Example 1. The following counterexample shows that the condition Ex2''(c) is indeed necessary. Let $\A$ be the additive category whose objects are morphisms of vector spaces $f\:V''\rarrow V'$ endowed with a subspace $V\sub\Im f$ and let $\T_\A$ be the class of all triples for which Ex1 holds. One can check that this class of admissible triples satisfies Ex0, Ex2''(a-b) and Ex3, but not Ex2''(c)."
That was the context in which my counterexample to what later became known as Raikov's conjecture was invented back in the '90s. In 2010 I took this example from the old file and put it into the new paper, which was subsequently accepted by MMJ.
So I studied both what (as I later learned) people call quasi-abelian and semi-abelian categories, and also maximal exact category structures on weakly idempotent-complete additive categories, back in the '90s, independently of any work by other people on these topics. And I did not publish my findings until the 2011 paper (which appeared as an arXiv preprint in 2010).
My motivation was to understand exact categories well enough to prepare myself to study the exact categories of mixed Artin-Tate motives (and motivic sheaves) with finite coefficients. These form pretty complicated exact categories, and I thought I needed to develop intuition on simpler examples and counterexamples first.
Чувствую себя каким-то реликтом века назад ушедшей в прошлое эпохи. Когда можно было так работать. Сначала 15 лет думать, потом публиковать так, чтобы никто не заметил, потом еще 10 лет думать, потом начинать людям рассказывать.
