Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.
Concept
Symmetric monoidal category
Résumé
In , a branch of mathematics, a symmetric monoidal category is a (i.e. a category in which a "tensor product" is defined) such that the tensor product is symmetric (i.e. is, in a certain strict sense, naturally isomorphic to for all objects and of the category). One of the prototypical examples of a symmetric monoidal category is the over some fixed field k, using the ordinary tensor product of vector spaces.
A symmetric monoidal category is a (C, ⊗, I) such that, for every pair A, B of objects in C, there is an isomorphism called the swap map that is natural in both A and B and such that the following diagrams commute:
The unit coherence:
The associativity coherence:
The inverse law:
In the diagrams above, a, l, and r are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.
Some examples and non-examples of symmetric monoidal categories:
The . The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
The . Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
More generally, any category with finite products, that is, a , is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
The over a ring R is monoidal (using the ordinary tensor product of modules), but not necessarily symmetric. If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field.
Given a field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the standard tensor product of representations is used.
The categories (Ste,) and (Ste,) of stereotype spaces over are symmetric monoidal, and moreover, (Ste,) is a symmetric monoidal category with the internal hom-functor .
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.