tag:blogger.com,1999:blog-3776650590176552530.post238374617547040459..comments2024-03-26T04:14:09.947-07:00Comments on TeX-nical Stuff: Fun with Functorspqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-3776650590176552530.post-44834206880636521442009-08-04T13:23:02.012-07:002009-08-04T13:23:02.012-07:00(Addendum: that link I gave is now broken, after p...(Addendum: that link I gave is now broken, after playing around with the html for a few days I've got the table of contents working -- see the top of the page below the title -- and the basic plan is outlined in the <a href="http://texnicalstuff.blogspot.com/2009/08/introduction.html" rel="nofollow">Introduction</a>.)pqnelsonhttps://www.blogger.com/profile/12779680952736168655noreply@blogger.comtag:blogger.com,1999:blog-3776650590176552530.post-81661393011273620142009-08-02T18:56:23.506-07:002009-08-02T18:56:23.506-07:00After thinking about it for a bit, when I get to c...After thinking about it for a bit, when I get to commutative diagrams I think I'd prefer to use metapost then convert it to a png, since metapost gives me a lot of freedom. Plus conversion is trivial with the "convert" program on Linux :)pqnelsonhttps://www.blogger.com/profile/12779680952736168655noreply@blogger.comtag:blogger.com,1999:blog-3776650590176552530.post-81092195154963915492009-08-02T18:35:18.615-07:002009-08-02T18:35:18.615-07:00It probably shouldn't be as impressive as it m...It probably shouldn't be as impressive as it might seem, I'm doing research on the subject of category theory in quantum physics (hence my choice of outline <a href="http://texnicalstuff.blogspot.com/2009/08/road-map.html" rel="nofollow">here</a>!). <br /><br />I'd be in a bad state if I couldn't do all these in one day...<br /><br />The diagram you've doodled (the second one) merely reiterates one of the main points of category theory: given any two morphisms $f:x\to{y}$, $g:y\to{z}$, we can always compose them. We just insist that when they so happen to be restriction morphisms that they result in a restriction when composed.<br /><br />This is important in insisting "consistency on overlaps", the sort of locality condition for presheaves. It then allows us to do geometry-type stuff with categories :)pqnelsonhttps://www.blogger.com/profile/12779680952736168655noreply@blogger.comtag:blogger.com,1999:blog-3776650590176552530.post-37060872835429044382009-08-02T18:24:12.175-07:002009-08-02T18:24:12.175-07:00Sorry there is some typo up there
\[\usepackage[al...Sorry there is some typo up there<br />\[\usepackage[all]{xy}\xymatrix{\ar@/^1.5pc/[rr]^{\mathrm{res}_{u,w}}F(u)\ar[r]_{\mathrm{res}_{u,v}}&F(v)\ar[r]_{\mathrm{res}_{v,w}}&F(w)}\]watchmathhttps://www.blogger.com/profile/13969830482192487872noreply@blogger.comtag:blogger.com,1999:blog-3776650590176552530.post-37463906987547448042009-08-02T18:17:59.297-07:002009-08-02T18:17:59.297-07:00Wow you write all of these in one day?
Just want t...Wow you write all of these in one day?<br />Just want to let you know that you can also use xypic package (it's been long time I didn't touch category theory :) )<br />Your restriction map can be described by this commutative diagram.<br />\[\usepackage[all]{xy}\xymatrix{\ar@/^1.5pc/[rr]^{\mathrm{res}_{u,w}}F(u)\ar[r]_{\mathrm{res}_{u,v}}&F(v)\ar[r]_{\mathrm{res}_{u,v}}&F(w)}\]watchmathhttps://www.blogger.com/profile/13969830482192487872noreply@blogger.com