Now to define the six functors formalism we have a scheme , a closed subscheme and an open subscheme such that on the level of the underlying sets our big scheme is the disjoint union. In order to write this using tikz-cd, which is an excellent package to write commutative diagrams, a bit of effort is needed. We define the appropriate arrow styles by

\usetikzlibrary{decorations.markings}
\tikzset{
  closed/.style = {decoration = {markings, mark = at position 0.5 with { \node[transform shape, xscale = .8, yscale=.4] {/}; } }, postaction = {decorate} },
  open/.style = {decoration = {markings, mark = at position 0.5 with { \node[transform shape, scale = .7] {$\circ$}; } }, postaction = {decorate} }
}

and then we can produce the diagram by

\begin{tikzcd}
  Z \arrow[hook, closed]{r}{i} & X \arrow[hookleftarrow, open]{r}{j} & U
\end{tikzcd}

This results in

Remark that we have to write the arrow from to in the opposite direction, because otherwise the hook is upside down. I haven’t found a better solution for this. And it seems that in the font I’m using (the Bera family with Charter from mathdesign for mathematics) the circle is slightly above the center of the line. If one goes this far in writing a silly diagram this should be fixed too, but unfortunately I haven’t found a (clean) solution for this yet. In the default font it looks better by the way.

It should be possible to do this in plain math mode too, but you’ll have to create long hooked arrows etc. This is easier and more consistent I think.

The todonotes package is a great tool if you’re as disorganized as I am, whenever I have a random idea, or don’t feel like finishing something or don’t want to add the correct citation, etc. I add a \todo{} and postpone the work. Postponing is good.

And if you can postpone work in a fancy way it’s even better! So I’ve got 5 “categories” of notes, each with its own color and options. An example (click for pdf):

The interesting part is that I’m using the xparse package, to have optional commands. So I can choose between \addreference and \addreference{to that particular book for which I don't feel like creating a BibTeX entry right now}. It’s really easy, just put

\usepackage{xparse}

\DeclareDocumentCommand\addreference{g}{\todo[color = blue!30]{Add reference\IfNoValueF{#1}{: #1}}}
\DeclareDocumentCommand\checkthis{g}{\todo[color = red!50]{Check this\IfNoValueF{#1}{: #1}}}
\DeclareDocumentCommand\fixthis{g}{\todo[color = orange!50]{Fix this\IfNoValueF{#1}{: #1}}}
\DeclareDocumentCommand\expand{g}{\todo[color = green!50]{Expand\IfNoValueF{#1}{: #1}}}
\newcommand\removethis{\todo[color = yellow!50]{Remove this?}}

in your preamble when you have todonotes loaded. Adding the option textsize = small to your \usepackage command might be a good idea, and there is colorinlistoftodos if you want colored boxes in the list.

During the summer I created a new website for the Stacks project (my announcement, official announcement). At that moment we’ve toyed with the idea of making a full-fledged online view, but the lack of a good xypic parser made this impossible. And what would the Stacks project be without commutative diagrams? At that point XyJax was already around, but it was incapable of rendering commutative diagrams, and it was inactive. Yet a few months ago it became active again, and after a request the developer implemented commutative diagrams (thank you!).

Now every tag has by default the online view of a tag (code view is still available if you want). A tag’s TeX code is processed in PHP to get something that approximates the pdf, after which MathJax and XyJax are let loose to render all math. Besides that, everything is tightly linked together so if you’re viewing a section’s tag (e.g. stacks:tag/06U0) you can:

• click on numbered theorems to go to their specific tag page;
• use all cross-references between tags;
• jump back and forth between sections.

So point your browser to the browse page, pick your favourite chapter and section, and you can start reading the Stacks project without opening any pdf’s, just by clicking around through the entire project. Don’t forget there is a powerful search functionality too.

If you encounter any bugs, please contact me. I am aware that it isn’t working properly in Opera at the moment.

It is an open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define them.

A long time ago I wrote about the Stacks project, when I just discovered it. Back then I didn’t know what a scheme is let alone a stack. And right now I still don’t know what a stack is. But as you might guess my contribution is the new website, not some chapter on those mysterious algebraic stacks.

First of all there is a new location: stacks.math.columbia.edu. All old URL’s will redirect to the new location by the way. The highlights of the new version are:

1. tag lookup: all results are given a tag, which is a unique and permanent identifier, if someone refers to “tag 03EV” you can look up its statement
2. comments: it is possible to discuss tags, pointing out typos, mistakes, or ask questions about it, it is possible to use Markdown and LaTeX in comments
3. browse: you can browse the project online, choose a chapter, look at its contents and you can navigate sequentially from tag to tag on the tag lookup page
4. fulltext search: if you come across the limits of searching a pdf you can use the fulltext search, which allows searching for tags or sections containing keywords, with the option to include or exclude proofs and the use of wildcards, logical operators and whatnot

There is also the official announcement, which already suggests the beta nature of the new website. Lots of functionality has been written, and there might (and probably will) be bugs. In case you come across one, you can either mail the Stacks project, leave a comment on the official announcement or on this post.

For a course in computational group theory I have to write a report about an implementation in GAP, and this is absolutely impossible without the best possible syntax highlighting, right? So I have entertained myself with implementing a GAP lexer. Its implementation is available at github.com/pbelmans/gap-pygments-lexer.

I have supplied an automatic installer, as the README suggests it boils down to

1. git clone git://github.com/pbelmans/gap-pygments-lexer.git or download and extract the archive GitHub automatically creates at the project’s Downloads page
2. cd gap-pygments-lexer
3. sudo python setup.py install

I would like to thank Wim Thys for providing me with the names of all the built-in classes and functions, using a bit of sed magic.

If you feel inspired, there are several other languages that need an implementation:

• Maple
• Magma (but they are be working on it themselves apparently)
• PARI/GP (this one has the most intriguing syntax of all)

So in text mode you write \v{C}ech, and in case you wish to write down the -th Čech cohomology group of a topological space and the sheaf you use \check{\mathrm{H}}^n(X,\mathcal{F}). Notice the use of \mathrm{H} for the actual (co)homology object, you could/should do this too!

Don’t shoot the nitpicky typesetting aficionado please. And consider using a macro like \newcommand\HH{\mathrm{H}} for easier typing. And of course \newcommand\cHH{\check{\mathrm{H}}}.

Let this fact be known too.

So I have decided to implement a UGent beamer theme myself, adhering to the style guide and following the philosophy set out by Nico. The implementation is available at GitHub, by means of the ugent-beamer project. I have rewritten Nico’s documentation to match my personal view on the subject of beamer themes, the result is available as a separate download. In case you wish to use the theme and you’re not that git-savvy, you can download the archive at the Downloads page. It contains all the necessary files, including the documentation.

The first two slides from the example I have supplied are

Installing beamer themes can be quite a hassle if you’re not acquainted with the inner workings of MiKTeX or TeX Live, but it should be thoroughly explained in the documentation. If anything is not clear or something is wrong, please let me know. As for now, there are still several things I would like to do with the theme in the future, they are listed at the Issues page.

\makeatletter
\newcommand\href@footnote[2]{#2\footnote{\url{#1}}}
\DeclareRobustCommand{\href}{\hyper@normalise\href@footnote}
\makeatother

This code properly handles active characters in url’s. Which is exactly what I want :).

I’m not claiming any originality here, but you might find it useful.

Anyhow, in case you want to use my material (conceived as 4 weeks of 3 hours classes each), please contact me, but otherwise you’re free to use them for your own benefit. The actual material is available at the ACM Antwerp LaTeX course page (not complete yet), the code necessary to create it is available at github.com/pbelmans/Dutch-LaTeX-course (where else…), using pdflatex --shell-escape slides and building several times should do the trick if you’ve got Pygments with correct Pygments style and the correct beamer theme installed.

I realize this post has little or no value if you don’t speak Dutch, my apologies if this applies to you.