Autonomous categoryIn mathematics, an autonomous category is a where dual objects exist. A left (resp. right) autonomous category is a where every object has a left (resp. right) dual. An autonomous category is a monoidal category where every object has both a left and a right dual. is a synonym for autonomous category. In a , the existence of left duals is equivalent to the existence of right duals, categories of this kind are called (symmetric) .
Category of relationsIn mathematics, the Rel has the class of sets as and binary relations as . A morphism (or arrow) R : A → B in this category is a relation between the sets A and B, so R ⊆ A × B. The composition of two relations R: A → B and S: B → C is given by (a, c) ∈ S o R ⇔ for some b ∈ B, (a, b) ∈ R and (b, c) ∈ S. Rel has also been called the "category of correspondences of sets". The category Rel has the Set as a (wide) , where the arrow f : X → Y in Set corresponds to the relation F ⊆ X × Y defined by (x, y) ∈ F ⇔ f(x) = y.
Rigid categoryIn , a branch of mathematics, a rigid category is a where every object is rigid, that is, has a dual X* (the internal Hom [X, 1]) 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 (following Alexander Grothendieck) by Neantro Saavedra Rivano in his thesis on . There are at least two equivalent definitions of a rigidity.
Dual objectIn , a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for in arbitrary . 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.
Dagger compact categoryIn , 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 reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations (that is, ). They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply , which describe general topological quantum field theories, for n = 1 and k = 3.
Dagger symmetric monoidal categoryIn the mathematical field of , a dagger symmetric monoidal category is a that also possesses a . That is, this category comes equipped not only with a tensor product in the sense but also with a , which is used to describe unitary morphisms and self-adjoint morphisms in : abstract analogues of those found in FdHilb, the . This type of was introduced by Peter Selinger as an intermediate structure between and the that are used in categorical quantum mechanics, an area that now also considers dagger symmetric monoidal categories when dealing with infinite-dimensional quantum mechanical concepts.
Monoidal categoryIn mathematics, a monoidal category (or tensor category) is a equipped with a bifunctor that is associative up to a natural isomorphism, and an I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant s commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.
Closed monoidal categoryIn mathematics, especially in , a closed monoidal category (or a monoidal closed category) is a that is both a and a in such a way that the structures are compatible. A classic example is the , Set, where the monoidal product of sets and is the usual cartesian product , and the internal Hom is the set of functions from to . A non- example is the , K-Vect, over a field . Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another.
Symmetric monoidal categoryIn , a branch of mathematics, a symmetric monoidal category is a (i.e. a category in which a "tensor product" is defined) such that the tensor product is symmetric (i.e. is, in a certain strict sense, naturally isomorphic to for all objects and of the category). One of the prototypical examples of a symmetric monoidal category is the over some fixed field k, using the ordinary tensor product of vector spaces.
