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RE: Bifunctor page
Bob Coeke, a Professor of Quantum Foundations, Logics and Structures at Oxford, is adamant about the primary role that category theory should have in foundational physics. I found two works of his online and as a physicist they really speak to me:
His and Aleks Kissinger’s work-in-progress treatise on quantum diagrammatica
Introducing categories to the practicing physicist
“Bifunctorality” is introduced in the latter work on page 7 and I was curious if it was in line with the bifunctor definition presented on nLab for one and your general opinions of his/their works for two?
I’ll answer the first question: yes, it’s perfectly in line.
A monoidal product is such a bifunctor, ⊗:M×M→M. Actually, I just call it a functor; a bifunctor is nothing more than a functor with two arguments. Anyway, what this means is that it takes composites of morphisms in M×M to composites in M, i.e., given (f:a→a′,g:b→b′)∈Mor(M×M) and (f′:a′→a″,g′:b′→b″)∈Mor(M), we have
⊗((f′,g′)∘(f,g))=⊗(f′,g′)∘⊗(f,g)Composition in Mor(M×M) is componentwise, i.e., (f′,g′)∘(f,g) is defined to be (f′∘f,g′∘g). Thus the equation above may be rewritten (dropping ∘’s):
f′f⊗g′g=(f′⊗g′)(f⊗g)(*)In particular, taking the special case where a′=a and b′=b″ and f=1a and g′=1b′, we have f′f=f′ and g′g=g, and the equation (*) above becomes
f′⊗g=(f′⊗1b′)(1a⊗g)for all morphisms f′:a→a″ and g:b→b′. In simpler notation: for all morphisms f,g, we have f⊗g=(f⊗1b′)(1a⊗g) where b′=cod(g) and a=dom(f).
On the other hand, taking the case a′=a″ and b=b′ and f′=1a′ and g=1b, we have f′f=f and g′g=g′, and the equation (*) becomes
f⊗g′=(1a′⊗g′)(f⊗1b)for all morphisms f:a→a′ and g′:b→b″. In simpler notation: for all morphisms f,g, we have f⊗g=(1a′⊗g)(f⊗1b) where a′=cod(f) and b=dom(g).
Hence we have, for all morphisms f:a→a′ and g:b→b′, the equations
(1a′⊗g)(f⊗1b)=f⊗g=(f⊗1b′)(1a⊗g)and the equation between the first and third terms, (1a′⊗g)(f⊗1b)=(f⊗1b′)(1a⊗g), is what Coecke is calling the bifunctoriality condition.
It may appear at first that Coecke’s bifunctoriality is just a special case of what we should be calling bifunctoriality according to the nLab. However, it is well-known that this interchange condition (related to the Eckmann-Hilton argument) is the crucial one. In more detail: of course one requires that each unary operator a⊗−, i.e., 1a⊗− and −⊗b, i.e., −⊗1b, for a monoidal product ⊗, should be functorial. But, these are not enough for ⊗ to be a bifunctor in the nLab sense. But, but, when we add the bifunctoriality condition in the sense of Coecke (I call it the interchange equation) to functoriality in each single argument, we get bifunctoriality in the nLab sense.
These observations were essentially first made by Roger Godement in the late 1950’s. Closely related material can be found at strict 2-category, sesquicategory, and at this recent MO post.
Fantastic! So if I took your meaning correctly: a nLab, two argument functor with interchange equation between these variables is the same as a Coecke’s “Bifunctorality”, which further explains the “braided” string diagrams.
If so, thanks for clearing that up and I look forward to any other comments you or others might have on these works!
Not quite: it’s Coecke’s bifunctoriality condition (aka the interchange condition) plus functoriality in each of two separate arguments that is the same as the notion of bifunctor given in the nLab (which is just a functor jointly in two arguments).
There are multiple ways of understanding braiding. One of the more intriguing ones is actually a failure to have an interchange equation: an interchange equation is replaced by an interchange isomorphism! But this must take place in higher categorical dimension, corresponding to the fact that braid string diagrams do not exist in the plane, but in 3-space.
I am not familiar in detail with Coecke’s and Kissinger’s achievements, so there others would have to comment. But it’s generally clear that categories-based diagrammatic methods in mathematics and physics are very useful and important. I am optimistic that the time will come when physicists (not to mention mathematicians) become generally comfortable with higher categorical structures, and no longer feel antagonized by the “Kategorienpest”. :-)
I am optimistic that the time will come when physicists (not to mention mathematicians) become generally comfortable with higher categorical structures, and no longer feel antagonized by the “Kategorienpest”. :-)
Thank you so much for the followup! I share your optimism which is why your insights and clarifications on this work are important to me, that I may more confidently spread it’s message.
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