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Colimits
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal a bijection between the respective morphism s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some directed set I, is denoted by \varinjlim A_i . (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.) Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are also a special case of limits in category theory. Formal definition We will first give the definition for algebraic structures like groups and modules, and then the general definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * In t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or ''inductive limit'', and a ''limit'' becomes a colimit. Formal definition Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) be a directed poset (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a family of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f_ is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * In t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by mean of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Co-cone
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-cone
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual (category Theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category ''C''op. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about ''C'', then its dual statement is true about ''C''op. Also, if a statement is false about ''C'', then its dual has to be false about ''C''op. Given a concrete category ''C'', it is often the case that the opposite category ''C''op per se is abstract. ''C''op need not be a category that arises from mathematical practice. In this case, another category ''D'' is also termed to be in duality with ''C'' if ''D'' and ''C''op are e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Cones
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Cone
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone from ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone (category Theory)
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Definition Let ''F'' : ''J'' → ''C'' be a diagram in ''C''. Formally, a diagram is nothing more than a functor from ''J'' to ''C''. The change in terminology reflects the fact that we think of ''F'' as indexing a family of objects and morphisms in ''C''. The category ''J'' is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when ''J'' is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes ''J'' to be a span. ''J'' can also be taken to be the empty category, leading to the simplest cones. Let ''N'' be an object of ''C''. A cone fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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