Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Posts Tagged ‘Homological algebra’

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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What is cohomology?

Posted by Andreas Holmstrom on December 13, 2008

Cohomology (or homology) means different things to different people. The common theme of all notions of cohomology, is the idea of using algebraic invariants to study geometric objects. More precisely, a cohomology theory is a functor from a geometric category (for example CW complexes or schemes) to an algebraic category (for example abelian groups, vector spaces, or modules). This is an extremely powerful idea, as the algebraic objects are often easier to work with, so a problem in geometry can be solved by transferring it to algebra.

We will be a bit sloppy in that we won’t really distinguish between cohomology and homology in the discussion below. Homology usually refers to functors which are covariant, while cohomology refers to functors which are contravariant.

Most commonly, the word cohomology is used to refer to singular cohomology, one of the fundamental notions of algebraic topology. More generally, algebraic topology studies and makes use of generalized cohomology and homology theories, such as K-theory, complex cobordism, and stable homotopy groups. Good online references for these things include this book of May (pdf) and the books of Hatcher.

In mathematics as a whole, there are over 400 different notions of cohomology. The reason for this multitude of cohomologies seems to be that almost any interesting functor from geometry to algebra is referred to as a cohomology theory, regardless of its properties. One of the very few things that all cohomology theories seem to have in common, is the appearance of long exact sequences, which is one of the most important tools for doing actual computations. More generally, the power of cohomology comes from the use of homological algebra, see for example these lecture notes (pdf) of Schapira.

Most of the cohomology theories in mathematics seem to appear in algebraic and arithmetic geometry. Many of these have helped solve some of the major mathematical problems of the past century. I will come back with more posts discussing these in more detail.

Posted in Cohomology theories in algebraic geometry | Tagged: , , | 6 Comments »