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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