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On "Generic Figures and Their Glueings" (Reyes/Reyes/Zolfaghari)

In 2009 I tried to read the book "Generic Figures and Their Glueings" (Reyes/Reyes/Zolfaghari, 2004, Polimetrica), but somewhere around p.40 (?) I got stuck - I remember vaguely not being able to resolve some ambiguities in the notation - and I gave up...

Some mentions to the book,
and online extracts:
http://nlab.mathforge.org/nlab/show/topos  (brief mention)
http://po-start.com/reyes/wp-content/uploads/2006/12/samplebook.pdf
http://books.google.com/books/about/Generic_figures_and_their_glueings.html?id=Z5GcOhfwoD0C&redir_esc=y
http://books.google.com.br/books?id=Z5GcOhfwoD0C&printsec=frontcover&redir_esc=y

In march 2012 two messages by Vasili Galchin in the Categories mailing list

http://article.gmane.org/gmane.science.mathematics.categories/7232
http://article.gmane.org/gmane.science.mathematics.categories/7234

convinced me to scan some of my old notes and put them on the web - here:

http://angg.twu.net/genericfigures/eduardo_ochs_RRZ_notes_2009.djvu
http://angg.twu.net/genericfigures/eduardo_ochs_RRZ_notes_2009.pdf

In Feb 2012 I gave an introductory minicourse on Category Theory at UnB (Universidade de Brasília) in which I used finite categories a lot - and now that I have much more practice with them I think it is time to try reading RRZ's book again. One trick that I've been using is that each finite subset of N+iN generates a DAG in a canonical way - the (basic) arrows are the "black pawn's moves" between its points, as in:

.                                1+3i                           1    
       			         /   \                         / \   
 *       add coordinates        v     v        relabel        v   v  
* *     in N+iN and arrows    2i      2+2i   the vertices    2     3 
 *     (black pawn's moves)     \     /      ===========>     \   /  
 *     ===================>      v   v       in "reading       v v   
			          1+i          order"           4    
			           |                            |    
			           v                            v    
			           1                            5    

...and DAGs can be interpreted as finite posets, so I can use a very compact notation - the diagram with the five stars at the left above (that I can typeset nicely in LaTeX, see this example) to represent some finite categories, and this even gives me a canonical way to name the objects of such categories!

Some ideas in the RRZ book may become clearer if we start with specific examples instead of generic ones (see this for some techniques for starting with examples) and if we name the objects and arrows explicitly... Time to try!

If you are interested in participating and/or receiving updates, please send an e-mail to:

"Eduardo Ochs" <eduardoochs@gmail.com>
"Vasili I. Galchin" <vigalchin@gmail.com>

(2012mar20)