- The Chu-Construction allows to obtain a *-autonomous category from the data of a closed symmetric monoidal category and a dualizing element.
- The Cayley-Dickson-construction builds an algebra B = A + A with involution from the data of an algebra A with involution *. Applied to the field of real numbers it gives successively the field of complex numbers, then the skew-field of quaternions, then the non-associative algebra of octonions, etc.
Due to closeness of the expressions of multiplication m: B \otimes B -> B for the multiplicative unit B we believe that there is an intimate link between both notions.
Has such a link been described in a reference text ?
Bibliography:
https://en.wikipedia.org/wiki/Cayley-Dickson_construction
https://ncatlab.org/nlab/show/Chu+construction
https://www.math.mcgill.ca/barr/papers/chu-hist
https://www.researchgate.net/publication/256555351_Nonsymmetric_-autonomous_categories
http://www.tac.mta.ca/tac/volumes/10/17/10-17abs.html
Have a look at https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
The question should be made more precise I guess : The Chu-construction deals with symmetric and associative tensor products, whereas the Cayley-Dickson-construction seems to loose first symmetry then associativity etc.
In the use of properties I understand that The 2-Chu (Cat, set) has duality, which allows isomorphism. But in the case of the Cayley-Dickson constructions it points to monomorphism in the unital sense. From which I could understand that there is not a direct link, but a relation by opposition of properties in the constructions, even though I could understand that Chu is about symmetric tension products and association but as a second phase of sequencing of such construction.
Dear Campo Elías Flórez Pabón ,
I am still used to the initial Chu-construction. Could you give some explanations or references about 2-Chu (...) ?
In think to fix ideas and to understand what happens it would be interesting to look first at particular cases of the basic Chu-construction.
- The standard Cayley-Dickson-construction in case 1 gives the field of complex numbers.
- The standard Cayley-Dickson-construction in case 2 gives the skew-field of quaternions. It is no longer symmetric, right ?
What is the difference between the two constructions in case 1 and in case 2 ?
Dear Thomas Krantz the references for me is https://golem.ph.utexas.edu/category/2017/11/the_2chu_construction.html
Interesting to consider the object Set in the (2-) category Cat as a dualizing object for the Chu-construction. Cat has a very rich structure and it is may be not the most simple example to consider for finding an answer to the question.
The Cayley-Dickson construction is for abelian categories - we should have at least an addition or a canonical isomorphism from the product to the coproduct of two copies of the unit object A for the tensor product.
This is a bit out of context but I think it is related to the question though.
Recalling that a monoid M is a one-object category, an involution * of the monoid M an endofunctor of M such that **=Id, consider an algebra k with conjugation written also * for simplicity.
The elements of the algebra k[M] are the finite formal sums
a1 e1+a2 e2 + ... + an en where ai is in k and ei in M.
Multiplication is defined by (a e)(b f):= (ab) (ef) with a,b in k and e, f in M.
Then the algebra k[M] seems to have a conjugation (a e)* := a* e*.
Note that * might or not reverse the order of multiplication of the elements of k
and might or not be contravariant on M.
Note that if M is a group, then M has a nice contravariant involution mapping e to e-1.
As a monoid is associative k[M] will be associative if k is also associative.
Has this construction something to do though with Caley-Dickson?
After this little excursion, I focus again on Caley-Dickson :
Given a non necessarily commutative neither associative (R-)algebra A, I think that the algebra B can be seen as A[i] where the following algebraic relations are satisfied for a in A.
ii = -1
i* = -i
a i = i a*
(a i) i = -a
The first relation can be dropped because it follows from the fourth one.
There are non commutative variants of the Chu-construction that might be closer to our needs. I added to the references :
Article Nonsymmetric -autonomous categories
and
http://www.tac.mta.ca/tac/volumes/10/17/10-17abs.html
See in particular results 3.10-3.13.
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