A conjectural characterization of symmetric powers in symmetric monoidal -linear categories is given on this page. The description in terms of universal properties would be equivalent to a much longer one but much more constructive in algebraic terms. It would be similar to the characterization of products known as Fox theorem but where products are replaced by symmetric powers. We work in symmetric monoidal categories enriched over modules over a -algebra. The adaptation to CMon-enriched symmetric monoidal categories would involve symmetric and divided powers which are equal in this simpler setting. This conjecture seems to be true and if it is really the case, the proof is not trivial.
Conjecture
We suppose that the symmetric monoidal categories are strict monoidal categories.
Definition
We say that a symmetric monoidal category possesses the -th symmetric power () iff it possesses this coequalizer for every : and is a natural transformation.
Definition
A functorial special bicommutative graded bimonoid in a -enriched symmetric monoidal category is given by:
for every , ) is a special bicommutative graded bimonoid ie.:
(commutativity)
(cocommutativity)
(unitality)
(counitality)
(associativity)
(coassociativity)
if , then (compatibility multiplication/comultiplication)
(specialty)
Conjecture
Let be a -algebra. Let be a symmetric monoidal category enriched over -modules. Let be a family of endofunctors such that and . Then, there is a bijection between families of natural transformations which make symmetric powers and families of natural transformations which make a functorial special bicommutative graded bimonoid.
Towards a proof
symmetric power in symmetric monoidal categories enriched over modules over a -algebra
Proposition
Let be a -algebra and be a symmetric monoidal category enriched over -modules. Let and be an endofunctor. There is a bijection between natural transformations which make symmetric powers and pairs of natural transformations such that:
Proof
Suppose that we have a natural transformation which make symmetric powers. Define the natural transformation . For every permutation , we have by using a change of variables in the last step. We thus have this factorization for every : It is not difficult to show that is a natural transformation.
Moreover . We know that is an epimorphism and thus .
Suppose now that we have a pair of natural transformations such that:
Suppose that is a morphism such that for every . Then . Suppose now that we have a morphism such that . Then, . is thus the only morphism such that . Thus, makes symmetric powers.
With this equivalent definition, we have already replaced the universal property defining symmetric powers by two natural transformations which verify some equations. The work is now to show the equivalence between these two equations and the ones defining a functorial special bicommutative graded bimonoid.
From symmetric powers to creation/annihilation operators
Definition
Let be -algebra and a symmetric monoidal category enriched over -modules. Suppose that we have a family of endofunctors such that and . Creation/annihilation operators are defined as two families and such that:
and:
where must be read as . Be aware that “” depicts the multiplication by the scalar .
Proposition
Let be -algebra and a symmetric monoidal category enriched over -modules. Suppose that we have a family of endofunctors such that and . Suppose that we have a family of natural transformations which make symmetric powers. We define creation/annihilation operators (for to particles) by putting:
From symmetric powers to functorial special bicommutative graded bimonoid
Proposition
Let be -algebra and a symmetric monoidal category enriched over -modules. Suppose that we have a family of endofunctors such that and . Suppose that we have a family of natural transformations which make symmetric powers. We define a (bounded) functorial special bicommutative graded bimonoid by putting: