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A monoidal adjunction is an adjunction in mathematics between monoidal categories which respects the monoidal structure. [1] [2] [3]
Suppose that and are two monoidal categories. A monoidal adjunction between two lax monoidal functors
is an adjunction between the underlying functors, such that the natural transformations
Suppose that
is a lax monoidal functor such that the underlying functor has a right adjoint . This adjunction lifts to a monoidal adjunction ⊣ if and only if the lax monoidal functor is strong.
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property.
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.
In category theory, a branch of mathematics, a monad is a triple consisting of a functor T from a category to itself and two natural transformations that satisfy the conditions like associativity. For example, if are functors adjoint to each other, then together with determined by the adjoint relation is a monad.
In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere, although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some limits and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category".
In mathematics, especially in category theory, a closed monoidal category is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to the question: "Does every monad arise from an adjunction?" The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli.
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 interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories.
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 a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors
Suppose that and are two monoidal categories and
In category theory, a branch of mathematics, 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 other words, is equipped with coherence maps and satisfying certain properties, and the unit and multiplication are monoidal natural transformations. By monoidality of , the morphisms and are necessarily equal.
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor 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 finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the category having finite-dimensional vector spaces as objects and linear maps as morphisms, with tensor product as the monoidal structure. Another example is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure.
A ribbon Hopf algebra is a quasitriangular Hopf algebra which possess an invertible central element more commonly known as the ribbon element, such that the following conditions hold:
In category theory, a branch of mathematics, dagger compact categories first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations. They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories, for n = 1 and k = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).
In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair:
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* and a morphism 1 → X ⊗ X* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined by Neantro Saavedra Rivano in his thesis on Tannakian categories.
In mathematics, weak bialgebras are a generalization of bialgebras that are both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras are weak bialgebras together with a linear map S satisfying specific conditions; they are generalizations of Hopf algebras.
