Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?
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3$\begingroup$ Gluing is an ordinary colimit, so you can certainly think of it as "some kind of 2-colimit", if you really believe that helps you. $\endgroup$– Johannes EbertMar 10, 2012 at 16:09
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$\begingroup$ Johannes is right. What you might be thinking is, the category of principal bundles over a fixed base is a $2$-colimit over all covers of the base (or some cofinal subset) of the categories of principal bundles over that fixed base which trivialize over the given cover. $\endgroup$– David CarchediMar 13, 2012 at 12:45
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$\begingroup$ @Carchedi: I wasn't actually ;), but after Johannes comment I realised he was right but I still felt there was a 2-colimit involved somewhere but was struggling to make the statement clear. Thanks for clarifying. $\endgroup$– Mozibur UllahMar 14, 2012 at 12:30
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not my answer, but David Carchedi's answer in a comment:
'What you might be thinking is, the category of principal bundles over a fixed base is a 2-colimit over all covers of the base (or some cofinal subset) of the categories of principal bundles over that fixed base which trivialize over the given cover'