Re: String diagrams, adjunction and autonomous categories.

[Micah Blake McCurdy <micah.mccurdy@gmail.com>]

Hallo!

In an autonomous category, the functor B -o _ is "representable" in the
sense that it can be written as B* (x) _ for some object B*. If k is the
unit for (x), then there are maps "unit" : k ----> B* (x) B and "counit" : B
(x) B* ---> k which satisfy triangle identities, usually written graphically
as zig-zags which equal identities. The isomorphism you mention in your
first paragraph is obtained by appending the unit for A.

So, the bends in the wires are the same in both paragraphs.

Incidentally, although I'm not sure that I'm familiar with the work of Baez
to which you refer, I would imagine that the use of string diagrams to
describe units and counits in an autonomous category is considerably older,
at least as far back as "Planar Diagrams and Tensor Algebra" by Joyal and
Street (available on the website of the latter) from 1988.

Cheers,

Micah

On Sun, Aug 29, 2010 at 2:48 AM, David Leduc <david.leduc6@googlemail.com>wrote:

> As shown by Baez, in an autonomous category the isomorphism hom(A (X) B, C)
> = hom (B, A -o C), when drawn as a string diagram, is like the bending of
> the input wire A to make it an output.
>
> Now one can also draw string diagrams to represent the zigzag equations
> between the adjoint pair of functors  _ (X) B and B -o _.
>
> How does the latter diagram relate to the former one?
>
>

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