Let and be fibered categories over a topological space . A morphism of fibered categories consists of: 1. a functor for each open subset and 2. a natural isomorphism for each inclusion . It is required that these structures satisfy a compatibility condition with respect to the 's, namely, that for the inclusions , , the above diagram should commute.
A fibered category over a topological space consists of 1. a category for each open subset , 2. a functor for each inclusion , and 3. a natural isomorphismfor each pair of inclusions , . In addition, for any three composable inclusions , , and , there exists a natural commuting as shown above.Sometimes, the pair is used to denote a fibered category with more precision while the shorthand is sometimes used for , , .
A functor is said to be faithful if it is injective on maps. This does not necessarily imply injectivity on objects. For example, the forgetful functor from the category of groups to the category of sets is faithful, but it identifies non-isomorphic groups having the same underlying set. Conversely, a functor injective on objects need not be injective on maps. For example, a counterexample is the functor on the category of vector spaces which leaves every vector space unchanged and sends every map to the zero map.A functor which is injective both on objects and maps is sometimes called an embedding.
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 Abelian groups and, more generally, the category of modules over a ring. Abelian categories are widely used in algebra, algebraic geometry, and topology.Many of the same constructions that are found in categories of modules, such as kernels, exact sequences, and commutative diagrams are available in Abelian categories. A disadvantage that must be overcome is the fact that the objects in a category do not necessarily have elements that can be manipulated directly, so the traditional definitions do not work. As a result, methods must be developed that allow definition and manipulation of objects without the use of elements.As an example, consider the definition of the kernel of a morphism, which states that given , the kernel of is defined to be a morphism such that all morphisms..
A tensor category is strict if the maps , , and are always identities.A related notion is that of a tensor R-category.
A natural transformation is called unital if the leftmost diagram above commutes. Similarly, a natural transformation is called unital if the diagram on the right-hand side above commutes.Note that in these definitions, , , and are all objects in a tensor category , is the neutral (or identity) object in , and the juxtaposition is shorthand for the tensor product in . What's more, the subscripts attached to the transformations and denote the components of the functors (indexed with respect to the objects in ) in question.
In category theory, a tensor category consists of a category , an object of , a functor , and a natural isomorphism(1)(2)(3)where the data are subject to the following axioms: 1. Given four objects , , , and of , the top diagram above commutes. 2. Given two objects and of , the bottom diagram above commutes. In the above, is called the tensor product, is called the associator, is called the right unit, and is called the left unit of the tensor category. The object is referred to as the neutral element or the identity of the tensor product.If the maps , , and are always identities, the tensor category in question is said to be strict.A related notion is that of a tensor R-category.
Let be functors between categories and . A natural transformation from to consists of a family of morphisms in which are indexed by the objects of so that, for each morphism between objects in , the equalityholds. The elements are called the components of the natural transformation.If all the components are isomorphisms in , then is called a natural isomorphism between and . In this case, one writes .
A natural transformation between functors of categories and is said to be a natural isomorphism if each of the components is an isomorphism in .
An equalizer of a pair of maps in a category is a map such that 1. , where denotes composition. 2. For any other map with the same property, there is exactly one map such that i.e., one has the above commutative diagram. It can be shown that the equalizer is a monomorphism.Moreover, it is unique up to isomorphism.In the category of sets, the equalizer is given by thesetand by the inclusion map of the subset in .The same construction is valid in the categories of additive groups, rings, modules, and vector spaces. For these, the kernel of a morphism can be viewed, in a more abstract categorical setting, as the equalizer of and the zero map.The dual notion is the coequalizer.