Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Posts Tagged ‘field with one element’

Semi-abelian categories

Posted by Andreas Holmstrom on November 1, 2009

The usual setting for doing homological algebra is abelian categories. However, many of the things one can do in abelian categories also make sense in more general settings. For example, the category of groups is not abelian, but one can still make sense of exact sequences, diagram lemmas, and so on.

A more general framework for doing homological algebra, which I first learnt about from Julia Goedecke, is given by the notion of semi-abelian categories. Some examples of semi-abelian categories are: groups, compact Hausdorff spaces, crossed modules, Lie algebras, any abelian category, and any category of algebras over a reduced operad (although I am not sure what it means for an operad to be reduced).

A very nice introduction and survey of semi-abelian categories can be found in the recent article of Hartl and Loiseau, on the arXiv. Other references include the nLab page and the thesis of Van der Linden.

The category of monoids is unfortunately not semi-abelian, but there was an interesting discussion on Math Overflow recently about making sense of homological algebra in the category of commutative monoids, which is interesting when trying to do algebraic geometry over the field with one element.

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Workshop on F1-geometry next month

Posted by Andreas Holmstrom on October 21, 2009

There will be a workshop on F1-geometry in Granada, 23-25 Nov. This has not been advertised much before, so spread the word. They plan a number of introductory lectures on various aspects of geometry over the field with one element, and it all looks very exciting. As an extra bonus, you might also get the chance to see the legendary range of symmetry groups in the Alhambra.

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New survey on Durov’s approach to the field with one element

Posted by Andreas Holmstrom on August 29, 2009

As Thomas points out in a comment, Javier Fresan has posted a survey on the thesis of Durov. Although Durov’s thesis is fairly readable, it is almost 600 pages long, so Fresan’s survey is a very welcome addition to the F1 literature.

For more on the field with one element, see F_un Mathematics. Durov’s work has also been discussed earlier at the n-category cafe and the Secret Blogging Seminar.

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Events in the coming year

Posted by Andreas Holmstrom on August 18, 2009

Over the past few weeks I have been updating the Events page, and there is now a long list of interesting conferences, workshops and summer schools taking place in the coming year. I wish there was a more sensible way of finding and listing events, like a “conference section” on the arXiv, following the subject classification used for preprints. The current way of doing things, with random mathematicians posting random lists of events on various webpages seems very inefficient for everyone involved. Anyway, if you share some of my interests then you are likely to find something useful in the updated list. Note that if you are young there is often funding available from the conference organisers if you apply early.

Some highlights:

See the full list for much more…

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