All these fundamental groups!

Tuesday, April 02nd, 2013 | Author:

There are a lot of fundamental groups floating around in mathematics. This is an attempt to collect some of the most popular and sketch their relations to each other.

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Homotopy Theory of Proofs?

Monday, March 25th, 2013 | Author:

Diagram proving the properties of a simplicial group covering

This question is not suitable for MathOverflow, as it is inherently vague.

In short:
Is there a Homotopy Theory of Proofs?

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Divisorial Jungle

Thursday, November 29th, 2012 | Author:

I'd like to compile a short list of definitions of Weil and Cartier Divisors, Line Bundles and Invertible Sheaves, Class Groups and Picard Groups, Cohomology, (higher) Chow Groups and K-theory for algebraic schemes and their relations. I intentionally omit proofs, but there are some ideas. I couldn't resist to jot down some properties of the objects which are important to me (homotopy invariance, existence of pullbacks and pushforwards).

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Model structures on simplicial presheaves

Friday, November 23rd, 2012 | Author:

This is a very short notice to memorize some of the various model structures on simplicial presheaves in a systematic way.

[UPDATE 2013-03-06] I gave a talk in our working group seminar about model structures on simplicial presheaves, homotopy sheaves and h-principles [/UPDATE]

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Cellular objects in the motivic model category

Friday, November 16th, 2012 | Author:

This article explains the notion of cellular objects in the motivic model category, which are like CW complexes for the algebraic/motivic world.

(last edit on 2014-04-08, added a remark on realizations and a Thom construction)

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Cellular objects: CW complexes

Wednesday, November 14th, 2012 | Author:

We will investigate the notion of cellular objects in a model category; today: the classical case of CW-complexes in the model category of topological spaces with Serre-fibrations as fibrations.

A CW complex is a certain kind of topological space, together with a CW structure, which is a description how to glue the space from spheres (or from affine spaces, if you prefer), called the cells. The acronym CW stands for "closure finite, weak topology", which I will explain soon. CW complexes are a class of spaces broader than simplicial complexes, and they are still combinatorial in nature.

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