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October 9, 2006

On n-Transport: Descent of the Universal Transition

Posted by Urs Schreiber

Last time # I talked about how the category of nn-paths (I consider n=1n=1 and n=2n=2 only) in a space XX and that of nn-paths in a regular surjection

(1)p:P n(Y)P n(X) p : P_n(Y) \to P_n(X)

give rise to the universal local transition P n(Y )P_n(Y^\bullet) of nn-transport # on XX; and that this is nothing but the category of nn-paths in YY which may “jump” between different lifts along pp.

Moreover, from any pp-local transition data of nn-transport (trivial transport on single patches, transitions gg of that on double intersections, transitions ff of these on triple intersections, and so on) one obtains a 2-transport

(2)(tra Y,g,f):P 2(Y )T. (\mathrm{tra}_Y,g,f)\; :\; P_2(Y^\bullet) \to T \,.

Clearly, this wants to descend to XX. The descent is manifest if

(3)P 2(Y )P 2(X). P_2(Y^\bullet) \simeq P_2(X) \,.

For general YY the constructions of this equivalence that I have managed to come up with (e.g. section 3. here) are a little unwieldy. But with a certain assumption on YY (which in common applications is always possible) it looks much better:

\,\,\,descent of the universal transition.

Posted at October 9, 2006 6:56 PM UTC

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