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...

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

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

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

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

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

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

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

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

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

mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object ⊥ {\displaystyle...

a commutative monoid; a Cartesian category with its finite products is an example of a symmetric monoidal category. For any objects X , Y ,  and  Z {\displaystyle...

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...

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...

more detail, this means that a category C is pre-abelian if: C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently...

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...

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...

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

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products...

bilinear; in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product (including the empty...

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked...

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

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

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

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...

In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the...

kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian categories are sometimes called AB2 categories, according...

In category theory, a branch of mathematics, a PROP is a symmetric strict monoidal category whose objects are the natural numbers n identified with the...

isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere...