Categorical background for A¹-homotopy theory (simplicial model categories)

Monday, December 07th, 2009 | Author:

I decided to post some background needed in order to understand Morel-Voevodsky's paper "A¹-homotopy theory". I explain some notions of simplicial sets, topoi, monoidal categories, enriched categories and simplicial model categories.

I tried to give many more references I found useful.
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Category: English, Walkthrough to A1-Homotopy Theory | 3 Comments

On Theories and Stories

Saturday, December 05th, 2009 | Author:

Many many times I talked about this topic in personal communication. I repeated myself and I developed my presentation of arguments over the years. The development will continue, but now I present you the current status. I hope you enjoy reading this and give some comments!
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Category: English, Not Mathematics | 2 Comments

Found nice introductory paper on cryptography & complexity

Tuesday, November 17th, 2009 | Author:

I just found this very nice paper on cryptography & complexity theory on the arXiv:

Jörg Rothe: "Some Facets of Complexity Theory and Cryptography: A Five-Lectures Tutorial"
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Category: English | Comments off

Haar measure in different settings

Sunday, November 15th, 2009 | Author:

I recently learned how to build a Haar measure on every locally compact group. It's a fact there is only one (up to positive scalar multiple) Haar measure on a locally compact group, and it's easy to see that Lie groups (which includes algebraic and finite groups) and all compact groups are locally compact, so they have a unique (up to scalar multiple) Haar measure, too.
But the Haar measure can be defined much easier for Lie groups, and it's even simpler for finite groups. I wanted to study the relation more directly than by the uniqueness proof one sees in the literature.
This text is intended to be read by anyone who is familiar with the notion of groups and measures. Maybe you will want to consult Wikipedia along the lines - I have included some links.

I give first a precise definition of Haar measure and a state its uniqueness on locally compact groups, then I compare the different types of topological groups I want to investigate, along with valid definitions of Haar measure.
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Category: English, Mathematics | 2 Comments

Gute Mathematik Skripten und Bücher

Wednesday, November 11th, 2009 | Author:

Ich mach mal eine Liste von den Grundlagen, und woher ich sie gelernt habe (=was ich als brauchbare Lektüre empfinde, also meine Lieblingsliteratur)
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Category: German, Mathematics | 3 Comments

Walk-through to Morel-Voevodsky A¹-homotopy theory part II (page 48, Lemma 1.1)

Wednesday, November 04th, 2009 | Author:

Here is the second part of my walk-through to Voevodskys A¹-homotopy theory:

On page 48, the first Lemma is shown. Without proof - so I will try to illuminate things a little bit by giving the proof. This lemma isn't used until section 3, so you can skip it, if you want to. I suggest not to do so, if you are intimidated by the diagram, because it isn't that hard, and it's a nice exercise to get the concepts in your head right.

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Category: English, Walkthrough to A1-Homotopy Theory | 5 Comments