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    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeOct 25th 2010

    I just googled

    +descent manifold “transition function”

    which led me to the Wikipedia article on connections, where we find a reference:

    The most abstract approach may be that suggested by Alexander Grothendieck, where a Grothendieck connection is seen as descent data from infinitesimal neighbourhoods of the diagonal; see (Osserman 2004).

    From there, I could not resist having a look at Osserman.

    From there, I noticed a slight resemblance to some things I’ve worked on.

    Warning: Entering shaky math zone. Help is appreciated.Apologies in advance for poor writing style, but I’m trying.

    When your space admits a finite set of complete orthogonal projections

    1= iπ iandπ iπ j=δ ijπ i,1 = \sum_i \pi_i\quad\text{and}\quad \pi_i\circ\pi_j = \delta_{i j}\pi_i,

    you can generate a free graded module of projections Ω\Omega. On this graded module, you can define a universal “graph operator” given by the innocuous looking

    G=11G = 1\otimes 1

    which can be expanded into projections via

    G= i,jπ iπ j.G = \sum_{i,j} \pi_i\otimes\pi_j.

    On certain special spaces (called “diamonds”), we can define a nilpotent map d:Ω nΩ n+1d:\Omega^n\to\Omega^{n+1} via the graded commutator with the graph operator

    dα[G,α]d\alpha \coloneqq [G,\alpha]

    for any αΩ\alpha\in\Omega. Note, in particular, that for fΩ 0f\in\Omega^0, we have

    df=[G,f]=1ff1.d f = [G,f] = 1\otimes f - f\otimes 1.

    This is in the kernel of a multiplication map m:Ω 1Ω 0m:\Omega^1\to\Omega^0 given by

    m(fg)=fg.m(f\otimes g) = f g.

    Now, if we interpret π iπ j\pi_i\otimes\pi_j as a directed edge π iπ j\pi_i\to\pi_j (hence the name “graph operator” above), then the universal graph operator corresponds to a complete graph. The differential graded algebra corresponding to this universal graph operator is a universal differential envelope.

    Any other differential graded algebra of projections can be obtained from this one by setting some of the π iπ j\pi_i\otimes\pi_j to zero and taking appropriate quotients.

    I am tempted to associate the process of setting some directed edges to zero with the process of defining the neighborhood of the diagonal. In other words, setting an element of the diagonal to zero means it is not in the neighborhood. Does that make any sense?

    PS: Since Ω 0\Omega^0 is generated by projections, we have

    f= if iπ i.f = \sum_i f_i \pi_i.

    Therefore, dfd f may be expressed in terms of projections via

    df=1ff1= i,j(f jf i)π iπ j.d f = 1\otimes f - f\otimes 1 = \sum_{i,j} (f_j - f_i) \pi_i \otimes \pi_j.
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 25th 2010
    • (edited Oct 25th 2010)

    Just a quick comment (I did not carefully try to understand Eric’s question yet). nnlab also has a Grothendieck connection entry. The descent data in question are usually known as “costratification”. Intuitive introduction is in the book by Berthelot and Ogus quoted there, paragraph 2.