Get your own LaTeX-enabled wiki in the cloud with Instiki on Heroku

Wednesday, November 21st, 2012 | Author:

I guess you all know what a WikiWikiWeb (short: wiki) is, it's a website where you can easily add new pages and modify existing ones. MathOverflow is some kind of hybrid between Q&A and a wiki, since users with enough reputation can edit other people's questions and answers. MathOverflow made the Markdown syntax very popular, and people got used to using LaTeX online. Some of my readers surely know the nLab, a collaborative wiki on n-categorical math(ematical physics) and stuff. The nLab runs on a software called Instiki, which is a wiki written in Ruby (an intepreted language similar to Python, and somewhat similar to Lisp, Perl and JavaScript; which is often used for web applications like wikis). The good thing about Instiki is that it supports editing pages in Markdown syntax with embedded LaTeX, so it is able to support your personal knowledge management needs. In addition, Instiki is small (thus not many bugs are to be expected), fast and the code is quite readable; something I wouldn't say about MediaWiki, the software behind Wikipedia.

In this post, I will tell you how to run your own wiki like the nLab. [UPDATED 2013-01-07; easier fix]

Category: English, Mathematics, Not Mathematics | 2 Comments

What's a point of this?

Monday, November 19th, 2012 | Author:

I recently came across a paper using a "universal domain" to discuss "generic points" of a variety, using Weil's foundations of algebraic geometry instead of Grothendieck's. First I had to learn that stuff, then I wanted to translate it. This lead to a more systematic study of what it means to be a point of a variety or scheme, in the various different definitions.

So in this post I will explain closed points, generic points, points in general position, schematic points, generalized points, rational points, geometric points, and in particular, which of these notions can be considered a particular case of another of these. I will try to give you a hint why one wants to generalize the ordinary (closed) points of a variety that much, to answer the question in the title: "What's the point of this?".

Category: English, Mathematics | 4 Comments

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)

Category: English, Mathematics | Comments off

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.

Category: English, Mathematics | Comments off

Model categories

Monday, November 12th, 2012 | Author:

This is supposed to be a short intuitive introduction to model categories.

Suppose you have a category $\mathcal{C}$ and some class of morphisms $W$ which behave somewhat like isomorphisms (for example: Chain complexes and Quasi-isomorphisms, or topological spaces and homotopy equivalences, or simplicial sets and weak homotopy equivalences ...). We will call this class "weak equivalences". Then you can look at the localized category $[W^{-1}]\mathcal{C}$, where the morphisms in $W$ are made invertible. If you're lucky, not all objects are isomorphic to each other, and if you're really lucky, you can compute something.

But, as it turns out, usually you don't work with the localized category abstractly, but by some explicit construction of some special case (say, Verdier localization of triangulated categories in the homological setting or explicit homotopies in the topological setting).

Model categories (and its cousins, weak factorization systems, categories of (co)fibrant objects, homotopical categories, etc.) provide a framework to compute stuff in $[W^{-1}]\mathcal{C}$.

[UPDATE 2013-03-06] I gave a 30-Minute talk about model categories, with very little content. [/UPDATE]

Category: English, Mathematics | 2 Comments

Homotopy limits

Tuesday, November 06th, 2012 | Author:

In this short posting, I want to give some intuitive idea on homotopy limits. Homotopy (co)limits appear whenever one has a notion of homotopy equivalence or weak equivalence between objects and one doesn't want to have constructions that distinguish between equivalent objects. The most prominent settings are, of course, classical homotopy theory and homological algebra. Although not necessary for the definition of homotopy (co)limits, I also talk about model categories.

Category: English, Mathematics | Comments off