Symmetric monoidal category information

In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ⊗ {\displaystyle...

In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle...

categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations...

mathematics, a commutativity constraint γ {\displaystyle \gamma } on a monoidal category C {\displaystyle {\mathcal {C}}} is a choice of isomorphism γ A ,...

In the mathematical field of category theory, a dagger symmetric monoidal category is a monoidal category ⟨ C , ⊗ , I ⟩ {\displaystyle \langle \mathbf...

(i.e., making the category symmetric monoidal or even symmetric closed monoidal, respectively).[citation needed] Enriched category theory thus encompasses...

is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure. A symmetric monoidal category ( C , ⊗ , I )...

category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback. A traced symmetric monoidal...

Derived category Triangulated category Model category 2-category Dagger symmetric monoidal category Dagger compact category Strongly ribbon category Closed...

{\mathcal {C}}}  : A symmetric monoidal functor is a braided monoidal functor whose domain and codomain are symmetric monoidal categories. The underlying functor...

of a commutative monoid; a category with finite coproducts is an example of a symmetric monoidal category. If the category has a zero object Z {\displaystyle...

consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which...

again, obeys certain coherence conditions (see symmetric monoidal category for details). A monoidal category is compact closed, if every object A ∈ C {\displaystyle...

bells and whistles in symmetric monoidal categories". arXiv:1908.02633 [math.CT]. Freyd, Peter J.; Scedrov, Andre (1990). Categories, Allegories. North Holland...

set, An (n + 1)-category is a category enriched over the category n-Cat. So a 1-category is just a (locally small) category. The monoidal structure of Set...

tensor product of modules ⊗, the category of modules is a symmetric monoidal category. A monoid object of the category of modules over a commutative ring...

the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both...

distributing over the other. A rig category is given by a category C {\displaystyle \mathbf {C} } equipped with: a symmetric monoidal structure ( C , ⊕ , O ) {\displaystyle...

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic...

preadditive category). The category of rings is a symmetric monoidal category with the tensor product of rings ⊗Z as the monoidal product and the ring of...

Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C...

into X. symmetric monoidal category A symmetric monoidal category is a monoidal category (i.e., a category with ⊗) that has maximally symmetric braiding...

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence...

tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. Since Hilbert spaces have inner products, one would like to...