Are the notions of dual category and dual object related? If not, are there any good reasons to use the word "dual" for both notions? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Sat, 4 Sep 2010, David Leduc wrote:
Are the notions of dual category and dual object related? If not, are there any good reasons to use the word "dual" for both notions?
Of course they're related. Under a categorical duality, the object in one category that corresponds to it in the other is the dual object. E.g. under the duality between sets and complete atomic boolean algebras, the object dual to to the set S is 2^S and the object dual to a CABA B is the set of atoms of B. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Sat, 4 Sep 2010, Toby Bartels wrote:
David Leduc wrote:
Are the notions of dual category and dual object related?
.......................................................
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}.
So you would say that complete atomic boolean algebras is just Set^{op}? Well I wouldn't. They are, of course, equivalent, but not the same. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
In the (bi)category Prof of categories and profunctors, the dual of an object is the dual category. Profunctors most certainly came later than the notions of categorical dual and dual objects (or at least their concrete counterparts, dual spaces), so this might just be a happy coincidence. Aleks On Sat, Sep 4, 2010 at 5:46 AM, David Leduc <david.leduc6@googlemail.com> wrote:
Are the notions of dual category and dual object related? If not, are there any good reasons to use the word "dual" for both notions?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 05/09/2010, at 5:41 AM, Aleks Kissinger wrote:
In the (bi)category Prof of categories and profunctors, the dual of an object is the dual category. Profunctors most certainly came later than the notions of categorical dual and dual objects (or at least their concrete counterparts, dual spaces), so this might just be a happy coincidence.
Very well put! I might add that an extra point needed is that Prof is a monoidal bicategory where the tensor product is the cartesian product of categories (it is not the cartesian product in Prof). And yes, Prof is compact, autonomous, rigid, whichever word you prefer, and the dual in Prof of a category A is A^{op}. In reading the literature, note that other names for Prof are Dist, Bimod and Mod. ==Ross
On Sat, Sep 4, 2010 at 5:46 AM, David Leduc <david.leduc6@googlemail.com
wrote: Are the notions of dual category and dual object related? If not, are there any good reasons to use the word "dual" for both notions?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Aleks Kissinger -
David Leduc -
Michael Barr -
Ross Street -
Toby Bartels