Re: Dual category and dual object

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

David Leduc wrote:

>Are the notions of dual category and dual object related?

The concept of dual vector space (which may be the historical original)
is an example of both, although in slightly different ways.
A vector space is an object in Vect, and its dual is a dual object;
taking a vector space to its dual extends to a contravariant functor,
which is the same thing as a functor on a dual category.

>If not, are there any good reasons to use the word "dual" for both notions?

Perhaps we should say "adjoint object" instead of "dual object".
Every monoidal category can be interpreted as a 2-category,
and a dual object in the monoidal category generalises to
an adjoint morphism in the 2-category.

On the other hand, we can say "opposite category" instead of "dual category",
especially since we denote the dual category of C by C^{op}.


--Toby


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