Concept
Monoidal category
Related concepts (28)
Closed monoidal category
In 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.
Enriched category
In , a branch of mathematics, an enriched category generalizes the idea of a by replacing hom-sets with objects from a general . It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an in some fixed monoidal category of "hom-objects".
Monoid (category theory)
In , a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a (C, ⊗, I) is an M together with two morphisms μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon and the unitor diagram commute. In the above notation, 1 is the identity morphism of M, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the Cop.
Hom functor
In mathematics, specifically in , hom-sets (i.e. sets of morphisms between ) give rise to important functors to the . These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Let C be a (i.e. a for which hom-classes are actually sets and not proper classes). For all objects A and B in C we define two functors to the as follows: {| class=wikitable |- ! Hom(A, –) : C → Set ! Hom(–, B) : C → Set |- | This is a covariant functor given by: Hom(A, –) maps each object X in C to the set of morphisms, Hom(A, X) Hom(A, –) maps each morphism f : X → Y to the function Hom(A, f) : Hom(A, X) → Hom(A, Y) given by for each g in Hom(A, X).
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group.
Category of abelian groups
In mathematics, the Ab has the abelian groups as and group homomorphisms as morphisms. This is the prototype of an : indeed, every can be embedded in Ab. The zero object of Ab is the trivial group {0} which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. Ab is a of Grp, the .
Compact closed category
In , 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 having finite-dimensional vector spaces as s and linear maps as s, with tensor product as the structure. Another example is , the category having sets as objects and relations as morphisms, with .
Strict 2-category
In , a strict 2-category is a with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category over Cat (the , with the structure given by ). The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou.
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. 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. Examples include free groups, tensor algebras, or free lattices. The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations).
Symmetric monoidal category
In , 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.