Dual category and dual object

Dual category and dual object

[David Leduc]

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?


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Re: Dual category and dual object

[Michael Barr]

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.


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Re: Dual category and dual object

[Toby Bartels]

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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Re: Dual category and dual object

[Aleks Kissinger]

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?
>
>


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Re: Dual category and dual object

[Michael Barr]

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.



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Re: Dual category and dual object

[Toby Bartels]

Michael Barr wrote:

>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 the same.

I would say both "The dual category of Set is Set^{op}."
and "CABA is a dual category of Set."
(and also "CABA is an opposite category of Set.").
Actually, I would probably do the grammar differently:
"CABA is a category dual [or "opposite"] to Set.".

Leduc's original question didn't have an article in it;
you seem to have interpreted is with "a" while I interpreted it with "the".
(We also interpreted "dual object" differently.)


--Toby


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Re: Dual category and dual object

[David Leduc]

Could you please spell out what is the unit in Prof (It is not Set, is
it?) and what are the units and counits for dual objects?

Thanks for your help.


On Sun, Sep 5, 2010, Ross Street <ross.street@mq.edu.au> wrote:
> 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.


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Re: Dual category and dual object

[Aleks Kissinger]

I originally thought it was Set, but this doesn't seem to make sense
using the cartesian product of categories as the monoidal product in
Prof. If this were the case, candidates for the unit and counit would
be the upper and lower stars of the hom functor.

However, let the 1-object category be the monoidal product. Then,
since 1 X C^op X C is isomorphic to C^op X C, we can regard the hom
functor of C as a profunctor 1 --|--> C^op X C,

Hom_C : 1 X C^op X C --> Set

and similarly, the hom functor of C^op as a profunctor C X C^op --|--> 1.

I saw this briefly mentioned here:
http://ncatlab.org/nlab/show/trace+of+a+category

Does anyone know if there is a more fleshed out treatment of this
somewhere? Also, do the "lifted" versions of the hom functor (upper
and lower star) serve some structural purpose in Prof?


Aleks


On Sat, Sep 4, 2010 at 8:50 PM, David Leduc <david.leduc6@googlemail.com> wrote:
> Could you please spell out what is the unit in Prof (It is not Set, is
> it?) and what are the units and counits for dual objects?
>
> Thanks for your help.
>
>
> On Sun, Sep 5, 2010, Ross Street <ross.street@mq.edu.au> wrote:
>> 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.
>


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