In category theory, **string diagrams** are a way of representing morphisms in monoidal categories, or more generally 2-cells in 2-categories.

The idea is to represent structures of dimension *d* by structures of dimension *2-d*, using Poincaré duality. Thus,

- an object is represented by a portion of plane,
- a 1-cell is represented by a vertical segment—called a
*string*—separating the plane in two (the right part corresponding to*A*and the left one to*B*), - a 2-cell is represented by an intersection of strings (the strings corresponding to
*f*above the link, the strings corresponding to*g*below the link).

The parallel composition of 2-cells corresponds to the horizontal juxtaposition of diagrams and the sequential composition to the vertical juxtaposition of diagrams.

Consider an adjunction between two categories and where is left adjoint of and the natural transformations and are respectively the unit and the counit. The string diagrams corresponding to these natural transformations are:

The string corresponding to the identity functor is drawn as a dotted line and can be omitted. The definition of an adjunction requires the following equalities:

The first one is depicted as

Morphisms in monoidal categories can also be drawn as string diagrams ^{ [1] } since a strict monoidal category can be seen as a 2-category with only one object (there will therefore be only one type of planar region) and Mac Lane's strictification theorem states that any monoidal category is monoidally equivalent to a strict one. The graphical language of string diagrams for monoidal categories may be extended to represent expressions in categories with other structure, such as braided monoidal categories, dagger categories,^{ [2] } etc. and is related to geometric presentations for braided monoidal categories ^{ [3] } and ribbon categories.^{ [4] } In quantum computing, there are several diagrammatic languages based on string diagrams for reasoning about linear maps between qubits, the most well-known of which is the ZX-calculus.

- TheCatsters (2007).
*String diagrams 1*(streamed video). Youtube. - String diagrams in
*nLab*

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. 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 antiautomorphism 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 mathematics, the idea of a **free object** is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure. It also has a formulation in terms of category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free lattices. Informally, a free object over a set *A* can be thought of as being a "generic" algebraic structure over *A*: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.

In category theory, a branch of mathematics, a **monad** is an endofunctor, together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

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 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 constraint, one must have for all pairs of objects .

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.

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

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 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.

Suppose that and are two monoidal categories. A **monoidal adjunction** between two lax monoidal functors

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.

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, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a **dualizable object**. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space *V*^{∗} doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property.

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, a **braided Hopf algebra** is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra *H*, particularly the Nichols algebra of a braided vector space in that category.

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, the **center** is a variant of the notion of the center of a monoid, group, or ring to a category.

In mathematics, in the theory of Hopf algebras, a **Hopf algebroid** is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf *k*-algebroids. If *k* is a field, a commutative *k*-algebroid is a cogroupoid object in the category of *k*-algebras; the category of such is hence dual to the category of groupoid *k*-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory. The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry. They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra *H* is a Hopf algebroid over its separable subalgebra *H ^{L}*. The antipode axioms have been changed by G. Böhm and K. Szlachányi in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions.

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.

In category theory, an abstract mathematical discipline, a **nodal decomposition** of a morphism is a representation of as a product , where is a strong epimorphism, a bimorphism, and a strong monomorphism.

- ↑ Joyal, André; Street, Ross (1991). "The geometry of tensor calculus, I" (PDF).
*Advances in Mathematics*.**88**(1): 55–112. doi: 10.1016/0001-8708(91)90003-P . ISSN 0001-8708. - ↑ Selinger, P. (2010). "A Survey of Graphical Languages for Monoidal Categories" (PDF). In Bob Coecke (ed.).
*New Structures for Physics*. Lecture Notes in Physics.**813**. Springer Berlin Heidelberg. pp. 289–355. arXiv: 0908.3347 . Bibcode:2009arXiv0908.3347S. doi:10.1007/978-3-642-12821-9_4. ISBN 978-3-642-12820-2. - ↑ Joyal, A.; Street, R. (1993). "Braided Tensor Categories".
*Advances in Mathematics*.**102**(1): 20–78. doi: 10.1006/aima.1993.1055 . ISSN 0001-8708. - ↑ Shum, Mei Chee (1994-04-11). "Tortile tensor categories".
*Journal of Pure and Applied Algebra*.**93**(1): 57–110. doi: 10.1016/0022-4049(92)00039-T . ISSN 0022-4049.

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