# Category: Monoidal categories

Dagger symmetric monoidal category
In the mathematical field of category theory, a dagger symmetric monoidal category is a monoidal category that also possesses a dagger structure. That is, this category comes equipped not only with a
Enriched category
In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observa
In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation tA,B : A ⊗ TB → T(A ⊗ B), called (tensorial) strength, such that the d
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibi
Monoidal functor
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of
6-j symbol
Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols, The summation is over all six mi allowed b
PROP (category theory)
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 finite sets and whose tensor product is given on
Rigid category
In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X* (the internal Hom [X, 1]) and a morphism 1 → X ⊗ X* satisfying
Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, betwee
Suppose that and are two monoidal categories. A monoidal adjunction between two lax monoidal functors and is an adjunction between the underlying functors, such that the natural transformations and ar
Frobenius category
In category theory, a branch of mathematics, a Frobenius category is an exact category with enough projectives and enough injectives, where the classes of projectives and injectives coincide. It is an
Dagger compact category
In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstru
Dual object
In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization,
Braided monoidal category
In mathematics, a commutativity constraint on a monoidal category is a choice of isomorphism for each pair of objects A and B which form a "natural family." In particular, to have a commutativity cons
In category theory, a monoidal monad is a monad on a monoidal category such that the functor is a lax monoidal functor and the natural transformations and are monoidal natural transformations. In othe
Monoid (category theory)
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms * μ: M ⊗
Yetter–Drinfeld category
In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.
Category of relations
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) R : A → B in this category is a relation between the sets A and B, so R ⊆ A ×
Monoidal category action
In algebra, an action of a monoidal category S on a category X is a functor such that there are natural isomorphisms and and those natural isomorphism satisfy the coherence conditions analogous to tho
Symmetric monoidal category
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" is defined) such that the tensor product is symmetric (i.
String diagram
String diagrams are a formal graphical language for representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When
Monoidal natural transformation
Suppose that and are two monoidal categories and and are two lax monoidal functors between those categories. A monoidal natural transformation between those functors is a natural transformation betwee
Compact closed category
In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a
Closed monoidal category
In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in such a way that the st
Rig category
In category theory, a rig category (also known as bimonoidal category or 2-rig) is a category equipped with two monoidal structures, one distributing over the other.
Tannakian formalism
In mathematics, a 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 is to approximate, in so
Traced monoidal category
In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback. A traced symmetric monoidal category is a symmetric monoidal categor
Categorical quantum mechanics
Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects
Ribbon category
In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category.
Center (category theory)
In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the notion of the center of a monoid, group, or ring
Cartesian monoidal category
In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any ca
Bialgebra
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made
Yang–Baxter equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, parti
Fiber functor
No description available.
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which g
Monoidal category
In mathematics, a monoidal category (or tensor category) is a category equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity
Weak inverse
In mathematics, the term weak inverse is used with several meanings.
Category of finite-dimensional Hilbert spaces
In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms.
*-autonomous category
In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object . The concept is also referred to as Grothendieck—Verdier ca
Autonomous category
In mathematics, an autonomous category is a monoidal category where dual objects exist.