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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]