From 3f687d41b98bae3341b36192d60eeb34a0f2f122 Mon Sep 17 00:00:00 2001 From: Gregor Kleen Date: Thu, 12 Mar 2015 13:33:08 +0000 Subject: Fork from math.kleen.org --- posts/blog-architecture-2.md | 33 -------------- posts/blog-architecture.md | 47 -------------------- posts/hello-world.md | 12 ------ posts/torsors.md | 100 ------------------------------------------- 4 files changed, 192 deletions(-) delete mode 100644 posts/blog-architecture-2.md delete mode 100644 posts/blog-architecture.md delete mode 100644 posts/hello-world.md delete mode 100644 posts/torsors.md (limited to 'posts') diff --git a/posts/blog-architecture-2.md b/posts/blog-architecture-2.md deleted file mode 100644 index 1e6ff43..0000000 --- a/posts/blog-architecture-2.md +++ /dev/null @@ -1,33 +0,0 @@ -% More on the Architecture of math.kleen.org - -It's been more than a month since the last post here. Sorry about that. But I -have now implemented a nice and fancy new feature for math.kleen.org, namely -theorem environments: - - -This could be your theorem here! - - - -Some theorems come with corollaries. - - -How these work ties in quite well with describing how LaTeX formulas are -converted to SVG for display in your browser. For example: -$$ -\int_{\partial M}\omega = \int_M\dd\omega -$$ - -Both these features are accomplished via extensive processing with -[Pandoc](http://en.wikipedia.org/wiki/Pandoc). For converting LaTeX snippets -into SVG, I have a little Haskell program that takes Pandocs native format, -extracts all math snippets and compiles them with latex and dvisvgm. This -program sits in the build subdirectory in -. - -For the theorem environment I patched the Markdown reader to convert mock HTML -tags like `` or `` into proper HTML. That way I can also retain the -possibility of converting posts into LaTeX and make PDFs out of them. - -I'm not sure whether I will continue documenting this software like that. But if -I ever write a redo replacement there will probably a few posts on the process. diff --git a/posts/blog-architecture.md b/posts/blog-architecture.md deleted file mode 100644 index 37cce77..0000000 --- a/posts/blog-architecture.md +++ /dev/null @@ -1,47 +0,0 @@ -% An Outline of the Architecture of math.kleen.org - -The main idea behind this system is that my site is supposed to consist of -*posts* which are organized into *lists of posts*. This mapping is not assumed -to be injective, i.e. a post may be a member of many lists. In fact, this is how -the list [All Posts](/lists/zz_all.html) is made: every post is supposed to be -linked into it. - -To keep order in this mess of data, I have decided to map it into the file -system. I have a folder `posts` which contains all posts and a folder `lists` -with all the lists. Every list is a folder with symbolic links to posts: - - blog - ├── lists - │ ├── all - │ │ ├── 001 -> ../../posts/hello-world.md - │ │ └── title - │ ├── blog - │ │ └── title - └── posts - └── hello-world.md - -As you can see, I have added a special file to each list folder named `title` -which contains the title of the list (as you might have guessed). - -Posts are written as [Markdown](http://en.wikipedia.org/wiki/Markdown) files and -converted to HTML with [Pandoc](http://en.wikipedia.org/wiki/Pandoc). Pandoc -handles this conversion almost perfectly, but I had one small issue with -it. Namely, I want to be able to write mathematics and hence translate -TeX--snippets into something that your browser can display. - -Pandoc has several builtin methods to do this, but most of them either rely on a -specialised TeX--parser or JavaScript. Both were deemed too ugly to use. So I -wrote a filter around Pandoc to extract TeX-snippets and compile them with my -regular LaTeX distribution into SVG. This seems to work quite nicely. - -The next issue was keeping this mess of posts and lists and Markdown files under -control. Traditionally, I would have used a Makefile for that but I wanted -something a little nicer this time. I turned to an old idea of -[Dan Bernstein's](http://cr.yp.to/djb.html): -[Redo](http://cr.yp.to/redo.html). There are several implementations of Redo out -there; eventually I plan to write my own, just as practice. For now, I use a -very minimal implementation in shell script from -. - -More on how exactly the conversion from Markdown to HTML and all the associated -ecosystem works will appear in a later post. diff --git a/posts/hello-world.md b/posts/hello-world.md deleted file mode 100644 index 5a886c1..0000000 --- a/posts/hello-world.md +++ /dev/null @@ -1,12 +0,0 @@ -% Hello, World! - -This is the inaugural post for this site. This "blog" is supposed to become -populated by a variety of posts pertaining to my research at USC, to personal -projects and (mathematicians, look away!) its own architecture. - -At the moment, I don't have much content; but hopefully this will change in due -time. In the first few posts, I think I will describe the custom software that I -put together to make this website happen; mostly for my benefit, so I have -documentation to fall back on once things break. - -This is all I have to say right now. Carry on! diff --git a/posts/torsors.md b/posts/torsors.md deleted file mode 100644 index a4cab97..0000000 --- a/posts/torsors.md +++ /dev/null @@ -1,100 +0,0 @@ -% Torsors and Their Classification I - -# Torsors in Grothendieck Toposes # - -Let $\ca C$ be some Grothendieck topos, that is a category of sheaves on some -Grothendieck topology. This text will look at some properties of *torsors* in -$\ca C$ and how they might be classified using cohomology and homotopy -theory. For this we will first need to define group objects. Torsors will then -be associated to those. - - A *group object* in a category $\ca C$ is an object $G\in\ca C$ such that -the associated Yoneda functor $\hom(\_,G)\colon \op{\ca C}\to\Set$ actually -takes values in the category $\Grp$ of groups. - -A trivial kind of example would be just a group considered as an object in -$\Set$. A more involved example would be a *group scheme* $G$ over some base -$S$. Such a thing is essentially defined as a group object in the category -$\sch[S]$. If we have any subcanonical topology on $\sch[S]$, then $G$ defines a -sheaf on the associated site and we obtain in this way a group object in the -corresponding topos on $S$. - -Let's now assume we have a group object $G$ in a Grothendieck topos $\ca -C$. Then we can define torsors over $G$ as follows: - - A *trivial $G$--torsor* is an object $X$ with a left $G$--action which is -isomorphic to $G$ itself with the action given by left multiplication. - - A *$G$--torsor* is an object $X\in\ca C$ with a left $G$--action which is -locally isomorphic to a trivial torsor; that is, there is an epimorphism $U\to -*$ such that $U\times X$ is a trivial torsor in $\ca C/U$ over $U \times G$. - - -I want to show that for any $G$--torsor $X$ according to this definition the -left action of $G$ on $X$ is free and transitive, that is the map -$$ f\colon G\times X \to X\times X $$ -given (on generalized elements) by $f(g, x) = (gx, x)$ is an isomorphism. This -is going to be some relatively elementary category theory but I think it's worth -writing it up. First a few facts about isomorphisms in toposes, they can be -found for example in Sheaves in Geometry and Logic.[^1] - - -Epimorphisms in a topos are stable under pullback. - - -In a topos every morphism $f\colon X\to Y$ has a functorial factorization $f = -m\circ e$ with $m$ a monomorphism and $e$ and epimorphism. - - -A morphism $f$ is an isomorphism if and only if $f$ is both monic and epic. - - -Now, let $f\colon A\to B$ be a *local monomorphism*, i.e. there is an epimorphism -$U\to *$ such that the pullback $f\times U$ of $f$ to $U$ is a -monomorphism. Then, since epimorphisms are stable under pullback, it follows that -in the commutative square -$$ -\begin{tikzcd} -A\times U \ar[into, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\ -A \ar[r, "f"'] & B -\end{tikzcd} -$$ -both vertical maps are epimorphisms. Now let $\varphi,\psi\colon T\to A$ be a pair of morphisms -such that $f\varphi = f\psi$. Then, denoting by $\varphi_U$ and $\psi_U$ the -pullbacks to $U$, we have $f_U\varphi_U = f_U\psi_U$. but $f_U$ is a -monomorphism by assumption, so $\varphi_U = \psi_U$. So we have a commutative -diagram -$$ -\begin{tikzcd} -T\times U \ar[r, "\varphi_U = \psi_U"] \ar[onto, d] & A\times U \ar[onto, d] \\ -T \ar[r, "\varphi", shift left] \ar[r, "\psi"', shift right] & A -\end{tikzcd} -$$ -in which the vertical maps are epimorphisms. It follows that $\varphi = -\psi$. - -Now, if $f$ is a local epimorphism, then again we have the diagram -$$ -\begin{tikzcd} -A\times U \ar[onto, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\ -A \ar[r, "f"'] & B -\end{tikzcd} -$$ -and it is immediate that $f$ is an epimorphism. In summary: - - -In a topos, any local epimorphism is an epimorphism, any local monomorphism is a -monomorphism, and any local isomorphism is an isomorphism. - - -Now, let's check that $G$--torsors $X$ as defined above are free and -transitive. Take an epimorphism $U\onto *$ such that $X\times U$ is trivial in -$\ca C/U$. The action of $G$ on itself by left multiplication is plainly free -and transitive, so in $\ca C/U$ we have the isomorphism -$$(G\times U)\times_U (X\times U) \iso (X\times U) -\times_U (X\times U)$$ -and $(G\times U)\times_U(X\times U) = (G\times X)\times U$ and $(X\times -U)\times_U (X\times U) = (X\times X)\times U$ because pullback preserves products. So, $G\times X\to -X\times X$ is a local isomorphism, hence an isomorphism. - -[^1]: Saunders Mac Lane, Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1994. ISBN: 0-387-97710-4 -- cgit v1.2.3