Concept
Adjoint functors
Related concepts (71)
Product (category theory)
In , the product of two (or more) in a is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
Equivalence of categories
In , a branch of abstract mathematics, an equivalence of categories is a relation between two that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned.
Functor
In mathematics, specifically , a functor is a mapping between . Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which is applied.
Tensor product
In mathematics, the tensor product of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map that maps a pair to an element of denoted An element of the form is called the tensor product of v and w. An element of is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span in the sense that every element of is a sum of elementary tensors.
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).
Functor category
In , a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, is another object in the category). Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.
Cartesian closed category
In , a is Cartesian closed if, roughly speaking, any morphism defined on a of two can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by , whose internal language, linear type systems, are suitable for both quantum and classical computation.
Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in , introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory.
Diagram (category theory)
In , a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.
Forgetful functor
In mathematics, in the area of , a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure.