summaryrefslogtreecommitdiff
path: root/posts
diff options
context:
space:
mode:
authorGregor Kleen <gkleen@praseodym.org>2015-03-12 13:33:08 +0000
committerGregor Kleen <gkleen@praseodym.org>2015-03-12 13:33:08 +0000
commit3f687d41b98bae3341b36192d60eeb34a0f2f122 (patch)
tree32bd4bdd79ecf82d7a96758a87ca208c88841ecd /posts
parent0303c4e1a700f6c5617f505a56cae88673a869da (diff)
downloaddirty-haskell.org-3f687d41b98bae3341b36192d60eeb34a0f2f122.tar
dirty-haskell.org-3f687d41b98bae3341b36192d60eeb34a0f2f122.tar.gz
dirty-haskell.org-3f687d41b98bae3341b36192d60eeb34a0f2f122.tar.bz2
dirty-haskell.org-3f687d41b98bae3341b36192d60eeb34a0f2f122.tar.xz
dirty-haskell.org-3f687d41b98bae3341b36192d60eeb34a0f2f122.zip
Fork from math.kleen.org
Diffstat (limited to 'posts')
-rw-r--r--posts/blog-architecture-2.md33
-rw-r--r--posts/blog-architecture.md47
-rw-r--r--posts/hello-world.md12
-rw-r--r--posts/torsors.md100
4 files changed, 0 insertions, 192 deletions
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 @@
1% More on the Architecture of math.kleen.org
2
3It's been more than a month since the last post here. Sorry about that. But I
4have now implemented a nice and fancy new feature for math.kleen.org, namely
5theorem environments:
6
7<thm>
8This could be your theorem here!
9</thm>
10
11<cor>
12Some theorems come with corollaries.
13</cor>
14
15How these work ties in quite well with describing how LaTeX formulas are
16converted to SVG for display in your browser. For example:
17$$
18\int_{\partial M}\omega = \int_M\dd\omega
19$$
20
21Both these features are accomplished via extensive processing with
22[Pandoc](http://en.wikipedia.org/wiki/Pandoc). For converting LaTeX snippets
23into SVG, I have a little Haskell program that takes Pandocs native format,
24extracts all math snippets and compiles them with latex and dvisvgm. This
25program sits in the build subdirectory in
26<https://github.com/vkleen/math.kleen.org>.
27
28For the theorem environment I patched the Markdown reader to convert mock HTML
29tags like `<thm>` or `<cor>` into proper HTML. That way I can also retain the
30possibility of converting posts into LaTeX and make PDFs out of them.
31
32I'm not sure whether I will continue documenting this software like that. But if
33I 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 @@
1% An Outline of the Architecture of math.kleen.org
2
3The main idea behind this system is that my site is supposed to consist of
4*posts* which are organized into *lists of posts*. This mapping is not assumed
5to be injective, i.e. a post may be a member of many lists. In fact, this is how
6the list [All Posts](/lists/zz_all.html) is made: every post is supposed to be
7linked into it.
8
9To keep order in this mess of data, I have decided to map it into the file
10system. I have a folder `posts` which contains all posts and a folder `lists`
11with all the lists. Every list is a folder with symbolic links to posts:
12
13 blog
14 ├── lists
15 │ ├── all
16 │ │ ├── 001 -> ../../posts/hello-world.md
17 │ │ └── title
18 │ ├── blog
19 │ │ └── title
20 └── posts
21 └── hello-world.md
22
23As you can see, I have added a special file to each list folder named `title`
24which contains the title of the list (as you might have guessed).
25
26Posts are written as [Markdown](http://en.wikipedia.org/wiki/Markdown) files and
27converted to HTML with [Pandoc](http://en.wikipedia.org/wiki/Pandoc). Pandoc
28handles this conversion almost perfectly, but I had one small issue with
29it. Namely, I want to be able to write mathematics and hence translate
30TeX--snippets into something that your browser can display.
31
32Pandoc has several builtin methods to do this, but most of them either rely on a
33specialised TeX--parser or JavaScript. Both were deemed too ugly to use. So I
34wrote a filter around Pandoc to extract TeX-snippets and compile them with my
35regular LaTeX distribution into SVG. This seems to work quite nicely.
36
37The next issue was keeping this mess of posts and lists and Markdown files under
38control. Traditionally, I would have used a Makefile for that but I wanted
39something a little nicer this time. I turned to an old idea of
40[Dan Bernstein's](http://cr.yp.to/djb.html):
41[Redo](http://cr.yp.to/redo.html). There are several implementations of Redo out
42there; eventually I plan to write my own, just as practice. For now, I use a
43very minimal implementation in shell script from
44<https://github.com/apenwarr/redo>.
45
46More on how exactly the conversion from Markdown to HTML and all the associated
47ecosystem 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 @@
1% Hello, World!
2
3This is the inaugural post for this site. This "blog" is supposed to become
4populated by a variety of posts pertaining to my research at USC, to personal
5projects and (mathematicians, look away!) its own architecture.
6
7At the moment, I don't have much content; but hopefully this will change in due
8time. In the first few posts, I think I will describe the custom software that I
9put together to make this website happen; mostly for my benefit, so I have
10documentation to fall back on once things break.
11
12This 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 @@
1% Torsors and Their Classification I
2
3# Torsors in Grothendieck Toposes #
4
5Let $\ca C$ be some Grothendieck topos, that is a category of sheaves on some
6Grothendieck topology. This text will look at some properties of *torsors* in
7$\ca C$ and how they might be classified using cohomology and homotopy
8theory. For this we will first need to define group objects. Torsors will then
9be associated to those.
10
11<defn> A *group object* in a category $\ca C$ is an object $G\in\ca C$ such that
12the associated Yoneda functor $\hom(\_,G)\colon \op{\ca C}\to\Set$ actually
13takes values in the category $\Grp$ of groups. </defn>
14
15A trivial kind of example would be just a group considered as an object in
16$\Set$. A more involved example would be a *group scheme* $G$ over some base
17$S$. Such a thing is essentially defined as a group object in the category
18$\sch[S]$. If we have any subcanonical topology on $\sch[S]$, then $G$ defines a
19sheaf on the associated site and we obtain in this way a group object in the
20corresponding topos on $S$.
21
22Let's now assume we have a group object $G$ in a Grothendieck topos $\ca
23C$. Then we can define torsors over $G$ as follows:
24
25<defn> A *trivial $G$--torsor* is an object $X$ with a left $G$--action which is
26isomorphic to $G$ itself with the action given by left multiplication. </defn>
27
28<defn> A *$G$--torsor* is an object $X\in\ca C$ with a left $G$--action which is
29locally isomorphic to a trivial torsor; that is, there is an epimorphism $U\to
30*$ such that $U\times X$ is a trivial torsor in $\ca C/U$ over $U \times G$.
31</defn>
32
33I want to show that for any $G$--torsor $X$ according to this definition the
34left action of $G$ on $X$ is free and transitive, that is the map
35$$ f\colon G\times X \to X\times X $$
36given (on generalized elements) by $f(g, x) = (gx, x)$ is an isomorphism. This
37is going to be some relatively elementary category theory but I think it's worth
38writing it up. First a few facts about isomorphisms in toposes, they can be
39found for example in Sheaves in Geometry and Logic.[^1]
40
41<lem>
42Epimorphisms in a topos are stable under pullback.
43</lem>
44<lem>
45In a topos every morphism $f\colon X\to Y$ has a functorial factorization $f =
46m\circ e$ with $m$ a monomorphism and $e$ and epimorphism.
47</lem>
48<lem>
49A morphism $f$ is an isomorphism if and only if $f$ is both monic and epic.
50</lem>
51
52Now, let $f\colon A\to B$ be a *local monomorphism*, i.e. there is an epimorphism
53$U\to *$ such that the pullback $f\times U$ of $f$ to $U$ is a
54monomorphism. Then, since epimorphisms are stable under pullback, it follows that
55in the commutative square
56$$
57\begin{tikzcd}
58A\times U \ar[into, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
59A \ar[r, "f"'] & B
60\end{tikzcd}
61$$
62both vertical maps are epimorphisms. Now let $\varphi,\psi\colon T\to A$ be a pair of morphisms
63such that $f\varphi = f\psi$. Then, denoting by $\varphi_U$ and $\psi_U$ the
64pullbacks to $U$, we have $f_U\varphi_U = f_U\psi_U$. but $f_U$ is a
65monomorphism by assumption, so $\varphi_U = \psi_U$. So we have a commutative
66diagram
67$$
68\begin{tikzcd}
69T\times U \ar[r, "\varphi_U = \psi_U"] \ar[onto, d] & A\times U \ar[onto, d] \\
70T \ar[r, "\varphi", shift left] \ar[r, "\psi"', shift right] & A
71\end{tikzcd}
72$$
73in which the vertical maps are epimorphisms. It follows that $\varphi =
74\psi$.
75
76Now, if $f$ is a local epimorphism, then again we have the diagram
77$$
78\begin{tikzcd}
79A\times U \ar[onto, r, "f\times U"] \ar[onto, d] & B\times U \ar[onto, d] \\
80A \ar[r, "f"'] & B
81\end{tikzcd}
82$$
83and it is immediate that $f$ is an epimorphism. In summary:
84
85<prop>
86In a topos, any local epimorphism is an epimorphism, any local monomorphism is a
87monomorphism, and any local isomorphism is an isomorphism.
88</prop>
89
90Now, let's check that $G$--torsors $X$ as defined above are free and
91transitive. Take an epimorphism $U\onto *$ such that $X\times U$ is trivial in
92$\ca C/U$. The action of $G$ on itself by left multiplication is plainly free
93and transitive, so in $\ca C/U$ we have the isomorphism
94$$(G\times U)\times_U (X\times U) \iso (X\times U)
95\times_U (X\times U)$$
96and $(G\times U)\times_U(X\times U) = (G\times X)\times U$ and $(X\times
97U)\times_U (X\times U) = (X\times X)\times U$ because pullback preserves products. So, $G\times X\to
98X\times X$ is a local isomorphism, hence an isomorphism.
99
100[^1]: Saunders Mac Lane, Ieke Moerdijk. Sheaves in geometry and logic. Springer, 1994. ISBN: 0-387-97710-4