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n-Forum: pro-C completion

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    • CommentRowNumber1.
    • CommentAuthorelif
    • CommentTimeAug 29th 2013
    In profinite algebraic homotopy page 111 Pro-C completions of a crossed modules and pro-C completions of the individual pieces of data involved are not always same. Can you give me a basic example about this?
    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeAug 29th 2013

    A good question. I thought the example on p.281 (in that version) might help, but I’m not sure it does. That section then goes on to look at Friedlander’s fibration example and that may also be helpful. Have a look at the Sl(2,5) example in more detail.

    • CommentRowNumber3.
    • CommentAuthorelif
    • CommentTimeAug 29th 2013
    I mentioned about profinite algebraic homotopy (shortened version). It has 239 pages. Are there any other version? İf you send me a link about it ,I will look at these example. On the other hand I want to ask you one more question? We can define other completion on algebras with its any ideal. (In commutative algebra this is called I-adic completion) So , can we apply this property on crossed modules of algebras ? And if it is defined, how can I apply same question in Pro-C completion? (page 111)
    Thanks for your concern...
    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeAug 30th 2013

    elif: There are longer versions, but in any case the examples are in section 5.5 of the version that you have.

    My advice to you is not to ask us the question, but to take apart what II-adic completion does in the commutative algebra case, and, importantly, what is it used for. Once you understand that well, you can try to see if an II-adic completion for crossed modules of commutative algebras/ cat^1-algebras is going to do something interesting. Again you need to find some examples of these things, say in polynomial rings and to produce some calculations (for yourself).

    When the work on pro-C completions of crossed modules was done, all the links between that stuff and higher dimensional algebra were less clear than they are now, so you need to look at that old stuff from the perspective of todays viewpoint, so as to make sure you are approaching it in a sufficiently general categorical way.

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