full subcategory - Article Index

Encyclopedia - Full And Faithful Functors: Encyclopedia Ii - Full And Faithful Functors - Examples
The forgetful functor U : Grp → Set is faithful but neither injective on objects or morphisms. This functor is not full as there a...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia - Adjoint Functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiqu...   » Read the article

Encyclopedia - Category Of Abelian Groups: Encyclopedia - Category Of Abelian Groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abeli...   » Read the article

Encyclopedia - Category Of Topological Spaces: Encyclopedia - Category Of Topological Spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose mor...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia - Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist an...   » Read the article

Encyclopedia - Compactly Generated Space: Encyclopedia Ii - Compactly Generated Space - Properties
Given any topological space X we can define a (possibly) finer topology on X which is compactly generated. Let {Kα} denote the family of...   » Read the article

Encyclopedia - Gluing Axiom: Encyclopedia Ii - Gluing Axiom - Removing Restrictions On C
To rephrase this definition in a way that will work in any category C that has sufficient structure, we note that we can write the object...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia Ii - Abelian Category - Definitions
A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia Ii - Adjoint Functors - Motivation
Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the ...   » Read the article

Encyclopedia - Category Of Topological Spaces: Encyclopedia Ii - Category Of Topological Spaces - Top Is A Concrete Category
Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) a...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia Ii - Abelian Category - Elementary Properties
Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero elem...   » Read the article

Encyclopedia - Gluing Axiom: Encyclopedia Ii - Gluing Axiom - Sheaves On A Basis Of Open Sets
In some categories, it is possible to construct a sheaf by specifying only some of its sections. Specifically, let X be a topological spa...   » Read the article

Encyclopedia - Gluing Axiom: Encyclopedia Ii - Gluing Axiom - The Logic Of C
The first needs of sheaf theory were for sheaves of abelian groups; so taking the category C as the category of abelian groups was only n...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia Ii - Adjoint Functors - Formal Definitions
A pair of adjoint functors between two categories C and D consists of two functors F : C → D and G : D → C and a natural is...   » Read the article

Encyclopedia - Category Of Topological Spaces: Encyclopedia Ii - Category Of Topological Spaces - Limits And Colimits
The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top. The forgetful functor U
...   » Read the article

Encyclopedia - Adjoint Functors: Encyclopedia Ii - Adjoint Functors - Examples
Free objects and forgetful functors. If F : Set → Grp is the functor assigning to each set X the free group over X, and if G ...   » Read the article

Encyclopedia - Gluing Axiom: Encyclopedia Ii - Gluing Axiom - Sheafification
How to turn a given presheaf P into a sheaf F? This is called sheafification, and there is a rough intuition of what one should do, at le...   » Read the article