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nLab > Latest Changes: New article: CABAs

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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeJan 30th 2010
    • (edited Jan 30th 2010)

    Does anybody know of a definition of CABA (which would not literally be complete atomic Boolean algebra) such that the theorem that the CABAs are (up to isomorphism of posets) precisely the power sets is constructively valid?

    If you do, you can put it here: CABA.

    • CommentRowNumber2.
    • CommentAuthorjoyal
    • CommentTimeJan 30th 2010

    I may have answered your question at

    CABA.

    • CommentRowNumber3.
    • CommentAuthorjoyal
    • CommentTimeJan 30th 2010

    Sorry, I made a mistake: I meant T_1-spaces, not T_0-spaces.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeJan 31st 2010

    Thanks, André! That answer makes sense to me.

    Do you have a reference? Or shall I ask on the categories mailing list?

    • CommentRowNumber5.
    • CommentAuthorjoyal
    • CommentTimeJan 31st 2010

    Actually, I maybe confused. If I remember it correctly, Paré studied the power object functor P:E-->E from an elementary topos E to itself.

    http://www.mscs.dal.ca/~pare/

    The functor is contravariant and self adjoint. Hence the square P^2=PP:E-->E has the structure of a monad. Paré had proved that the category of algebras of this monad is equivalent to E^o. In other words, the opposite category E^o is monadic over E. As an application, this was showing that E has finite colimits (the category E^o has finite limits since it is monadic over E and E has finite limits).

    About your problem. If A is an object of E, then the object P(A) is the frame of a discrete locale. A locale X is said to be discrete if the canonical map X--->1 and the diagonal X--->X times X are open. The category E is equivalent to the category of discrete locales in E.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeJan 31st 2010

    A locale X is said to be discrete if the canonical map X--->1 and the diagonal X--->X times X are open.

    OK, I think I knew that but I wasn't remembering that I knew it.

    It would be nice to have this in purely order-theoretic terms, to match the classical definition of CABA (and prove that they are classically equivalent). But I can work that out myself, and in principle it is done already. Thanks.

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