Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Archive for November, 2009

Motives in Paris and Regulators in Barcelona

Posted by Andreas Holmstrom on November 25, 2009

The Events page has been updated again. Some highlights:

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Blog silence because of Math Overflow…

Posted by Andreas Holmstrom on November 20, 2009

The last few weeks have been quite busy, and the spare moments that I would normally spend on blogging have been hi-jacked by Math Overflow. I wrote a few things there which I would normally have put on this blog, and since they might possibly be of interest to some blog readers, here are the links: Why are functional equations important, and What is the Yoga of Motives.

For quite a while, I have been trying (without much success) to understand finiteness properties for simplicial sheaves, and thanks to MO, I got an absolutely brilliant explanation from Denis-Charles Cisinski – something which simply could not have happened otherwise.  Lots of credit to MO (and to Cisinski)!

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Intersection theory at Rigorous Trivialities

Posted by Andreas Holmstrom on November 3, 2009

Charles Siegel at Rigorous Trivialities is aiming to blog about intersection theory every day of November, essentially creating a minor book in the process. The first two posts are out, one on Chow groups and one on Manipulating cycles. These posts look promising, and I am very much looking forward to the rest of the series!

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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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