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Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

December 25, 2013

The Long Grind of Writing a Book

Posted by Tom Leinster

I’m using the quiet of Christmas to finish writing a book, Basic Category Theory. It’s nothing revolutionary: just a short introduction to the subject, based on courses I’ve taught. But the process of book-writing is demanding and maddening enough that I wanted to take a moment to reflect on why that is — and why you hear authors complain so much.

Put another way, I’m taking a break from the tedium of writing a book to write about the tedium of writing a book. I hope it’s not tedious.

Posted at 3:28 PM UTC | Permalink | Followups (50)

December 21, 2013

Commuting Limits and Colimits over Groups

Posted by Tom Leinster

Limits commute with limits, and colimits commute with colimits, but limits and colimits don’t usually commute with each other — with some notable exceptions. The most famous of these is that in the category of sets, finite limits commute with filtered colimits.

Various other cases of limit-colimit commutation are known. There’s an nLab page listing some. But it seems that quite an easy case has been overlooked.

It came to light earlier this week, when I was visiting Cambridge. Peter Johnstone told me that he’d found a family of new limit-colimit commutations in the category of sets, I asked whether his result could be simplified in a certain way (to involve groups only), and we both realized that it could not only be simplified, but also generalized.

Here it is. Let GG and HH be finite groups whose orders are coprime. View them as one-object categories. Then GG-colimits commute with HH-limits in the category of sets.

Posted at 12:55 AM UTC | Permalink | Followups (10)

December 10, 2013

A Technical Innovation

Posted by Tom Leinster

Here’s a new feature of the Café, thanks to our benevolent host Jacques Distler. If you ever want to see how someone has created some mathematical expression on this blog, there’s an easy way to do it.

With Firefox, you simply double-click on the expression. Try it: A×B ABA \times B^A \to B or x mnx_{m n} or (1 2 3 4). \Biggl( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \Biggr). A window should pop up showing the TeX source.

With other browsers, I’m not so sure. Try double-clicking. If that doesn’t work, then, according to Jacques’s instructions, you “bring up the MathJax context-menu for the formula, and choose Show Math As \to Annotation \to TeX”. I don’t know how one brings up this menu. Does anyone else know? (Update: right-click in Chrome, Explorer and Opera, and control-click in Safari. Thanks to those who responded.)

Once you’ve made the TeX source appear, you can cut and paste to your heart’s content. Of course, most users here are fluent in LaTeX. But like most math-oriented websites, we use a variant of TeX that’s a little different from standard LaTeX, so this should turn out to be a helpful feature.

Posted at 1:41 AM UTC | Permalink | Followups (18)