full subcategory

The forgetful functor U : Grp → Set is faithful but neither injective on objects or morphisms. This functor is not full as there are functions between groups which are not group homomorphisms. More generally, for any concrete category the forgetful functor to Set is faithful (but usually not full). Let F : C → Set be the functor which maps every object in C to the empty set and every morphism to the empty function. Then F i ...

See also:

Full and faithful functors, Full and faithful functors - Examples

Read more here: » Full and faithful functors: Encyclopedia II - Full and faithful functors - Examples

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia - Adjoint functors

In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. The zero object of Ab is the trivial group {0} which consists only of its neutral element. Note that Ab is a full subcategory of Grp, the category of all ...

Read more here: » Category of abelian groups: Encyclopedia - Category of abelian groups

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms ...

Including:

• Category of topological spaces - Top is a concrete category
• Category of topological spaces - Limits and colimits
• Category of topological spaces - Other properties
• Category of topological spaces - Relationships to other categories

Read more here: » Category of topological spaces: Encyclopedia - Category of topological spaces

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. The motivating prototype example of an abelian category is the category of abelian groups, Ab. 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 a theorem of Pe ...

Including:

• Abelian category - Definitions
• Abelian category - Examples
• Abelian category - Elementary properties
• Abelian category - Related concepts
• Abelian category - History

Read more here: » Abelian category: Encyclopedia - Abelian category

Given any topological space X we can define a (possibly) finer topology on X which is compactly generated. Let {Kα} denote the family of compact subsets of X. We define the new topology on X by declaring a subset A to be closed iff A ∩ Kα is closed in Kα for each α. Denote this new space by Xc. One can show that the compact subsets of Xc and X coincide and the induced topologies are the same. It fol ...

See also:

Compactly generated space, Compactly generated space - Properties

Read more here: » Compactly generated space: Encyclopedia II - Compactly generated space - Properties

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 objects and morphisms involved in the definition above in a diagram which we will call (G), for "gluing": Here the first map is the product of the restriction maps resU,Ui,:F(U)→F(Ui) and each pair of arrows represents the two restrictions resUi,Ui∩Uj:USee also:

Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

Read more here: » Gluing axiom: Encyclopedia II - Gluing axiom - Removing restrictions on C

A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition: A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. A preadditive category is ...

See also:

Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

Read more here: » Abelian category: Encyclopedia II - Abelian category - Definitions

Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as Hom(F(X), Y< ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia II - Adjoint functors - Motivation

Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor U : Top → Set to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function. ...

See also:

Category of topological spaces, Category of topological spaces - Top is a concrete category, Category of topological spaces - Limits and colimits, Category of topological spaces - Other properties, Category of topological spaces - Relationships to other categories

Read more here: » Category of topological spaces: Encyclopedia II - Category of topological spaces - Top is a concrete category

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 element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, whil ...

See also:

Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

Read more here: » Abelian category: Encyclopedia II - Abelian category - Elementary properties

In some categories, it is possible to construct a sheaf by specifying only some of its sections. Specifically, let X be a topological space with basis {Bi}i∈I. We can define a category Top'X to be the full subcategory of TopX whose objects are the {Bi}. A B-sheaf on X with values in C is a contravariant functor F:Top'X→C which satisfies the gluing axiom for sets in Top'X. We would like to recover the va ...

See also:

Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

Read more here: » Gluing axiom: Encyclopedia II - Gluing axiom - Sheaves on a basis of open sets

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 natural. In applications to geometry, for example complex manifolds and algebraic geometry, the idea of a sheaf of local rings is central. This, however, is not quite the same thing; one speaks instead of a locally ringed space, because it is not true, except in trite cases, that such a sheaf is a functor into a category of local rings. It is the stalks of the sheaf that are local rings, ...

See also:

Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

Read more here: » Gluing axiom: Encyclopedia II - Gluing axiom - The logic of C

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 isomorphism φ : MorD(F–, –) → MorC(–, G–) consisting of bijections: φX,Y : MorD(F(X), Y) → Mor< ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia II - Adjoint functors - Formal definitions

The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top. The forgetful functor U : Top → Set has a left adjoint which equips a given set with the discrete topology and a right adjoint which equips a given set with the trivial topology. This implies that the functor U is both limit-preserving and colimit-preserving, i.e. limits in Top are ...

See also:

Category of topological spaces, Category of topological spaces - Top is a concrete category, Category of topological spaces - Limits and colimits, Category of topological spaces - Other properties, Category of topological spaces - Relationships to other categories

Read more here: » Category of topological spaces: Encyclopedia II - Category of topological spaces - Limits and colimits

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 : Grp → Set is the forgetful functor assigning to each group its underlying set, then the universal property of the free group shows that F is left adjoint to G. The unit of this adjoint pair is the embedding of a set X into the free group over X. In general, free constructions in mathematics tend to be left adjoints of forgetful functors. Free rings, free abelian groups, ...

See also:

Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia II - Adjoint functors - Examples

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 least for a presheaf of sets. One should introduce an equivalence relation, which makes equivalent data given by different covers which 'become' equivalent by refining the covers. One approach is therefore to go to the stalks and recover the sheaf space of the best poss ...

See also:

Gluing axiom, Gluing axiom - Removing restrictions on C, Gluing axiom - Sheaves on a basis of open sets, Gluing axiom - The logic of C, Gluing axiom - Sheafification, Gluing axiom - Other gluing axioms

Read more here: » Gluing axiom: Encyclopedia II - Gluing axiom - Sheafification