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A category consists of three things: a collection of objects, for each pair of objects a collection of morphisms (sometimes call "arrows") from one to another, and a binary ...

A mathematical structure A is said to be closed under an operation + if, whenever a and b are both elements of A, then so is a+b. A mathematical object taken together with ...

The branch of mathematics which formalizes a number of algebraic properties of collections of transformations between mathematical objects (such as binary relations, groups, ...

Let R be a commutative ring. A category C is called an R-category if the Hom-sets of C are R-modules.

A subset E of a topological space S is said to be of second category in S if E cannot be written as the countable union of subsets which are nowhere dense in S, i.e., if ...

A subset E of a topological space S is said to be of first category in S if E can be written as the countable union of subsets which are nowhere dense in S, i.e., if E is ...

An Abelian category is a category for which the constructions and techniques of homological algebra are available. The basic examples of such categories are the category of ...

An additive category is a category for which the morphism sets have the structure of Abelian groups. It satisfies some, but not all the properties of an Abelian category.

In category theory, a tensor category (C, tensor ,I,a,r,l) consists of a category C, an object I of C, a functor tensor :C×C->C, and a natural isomorphism a = a_(UVW):(U ...

Let R be a commutative ring. A tensor category (C, tensor ,I,a,r,l) is said to be a tensor R-category if C is an R-category and if the tensor product functor is an R-bilinear ...