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In mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms. If K is the field of real numbers, then the category is also known as Vec. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod, the category of left R-modules. K-Vect is an important example of an abelian category. In abstract algebra, a module is a generalization of a vector space. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...
Much of linear algebra concerns the description of K-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in K-Vect correspond exactly to the cardinal numbers, and that K-Vect is equivalent to the subcategory of K-Vect which has as its objects the free vector spaces Kn, where n is any cardinal number. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
In mathematics, the dimension theorem for vector spaces states that a vector space has a definite, well-defined number of dimensions. ...
An isomorphism class is a collection of mathematical objects isomorphic with a certain mathematical object. ...
Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ...
In mathematics, a subcategory S of a category C consists of subsets of the morphisms and of the objects of C, such that the subset X of morphisms is closed under composition in C, and the subset Y of objects contains the source and target of all the f in...
There is a forgetful functor from K-Vect to Ab, the category of abelian groups, which takes each vector space to its additive group. This can be composed with forgetful functors from Ab to yield other forgetful functors, most importantly one to Set. A forgetful functor is a type of functor in mathematics. ...
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...
In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...
K-Vect is a monoidal category with K (as a one dimensional vector space over K) as the identity and the tensor product as the monoidal product. In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). ...
In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...
See also
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