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There exist a structure on double categories due to R.Brown called a connection. The connection embodies in squares an isomorphism between the category of its vertical arrows and the category of its horizontal arrows and it allows to change boundaries of a square from one type to the other. It generates a double subcategory that is very interesting to me. I was wondering if anyone had thought of and found a way to describe this subcategory without the mention of "commutative squares" or the construction of the connection itself. I have the feeling that it is given by an adjunction but it seems to elude me.

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  • $\begingroup$ In the question you have written: "...It generates a double subcategory that is very interesting to me... " I would like to know if you can be more precise at this point. That is, what is the big double category to which you have made reference? $\endgroup$
    – Jesús
    Jan 13, 2011 at 1:50
  • $\begingroup$ The big double category in question is what is called the quintet category, it is constituted of all possible squares, not only the commutative ones). $\endgroup$
    – Dany Majard
    Jan 13, 2011 at 18:31

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Have you see http://www.tac.mta.ca/tac/volumes/1999/n7/5-07abs.html ?

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  • $\begingroup$ Yes Buschi, I know this paper but it doesn't answer my question sadly... $\endgroup$
    – Dany Majard
    Nov 9, 2010 at 20:02

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