Re: Dualities arising via pairs of schizophrenic objects

["Fred E.J. Linton" <fejlinton@usa.net>]

On Fri, 02 Dec 2011 08:35:30 AM EST, David Roberts
<david.roberts@adelaide.edu.au> wrote:
  
>  ... always thought it odd that even when one wants to accept
> category-theoretic
> foundations (e.g. ETCS or similar), then suddenly something like this
> comes along,
> where people start saying there is a thing which is an object of two
> different categories.

Ever since Eckmann-Hilton, and perhaps even before, the notion of an
object G in one category X bearing the structure of an object in some
concrete other category A (concrete via U: A -> Sets, say) has been 
clearly and unambiguously expressed as follows:

The hom functor X(-, G): X^op -> Sets is given a factorization thru' U.

If both X and A are concrete, it's perfectly plausible for an object
of X to bear the structure of an object in A, and vice versa, and a
brief peek at the example of 2 as BA w/ KT_2-space structure and as
KT_2-space with BA structure will make short work of understanding how
an object may be thought of as "inhabiting both categories at once":
indeed, it's that contravariant adjoint pair alone, between A and X,
that provides the duality in John Isbell's 1972 approach, where 
at most one of A and X need be concrete.

HTH. Cheers, -- Fred



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