Information about Diagonal Functor
In category theory, for any object a in any category C where the product a×a exists, there exists the diagonal morphism
satisfying
is the canonical projection morphism to the k-th component. The existence of this morphism is a consequence of the universal property which characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.
For concrete categories, the diagonal morphism can be simply described by its action on elements x of the object a. Namely, δa(x) = (x,x), the ordered pair formed from x. The reason for the name is that the graph of such a diagonal morphism is diagonal (whenever it makes sense), for example the graph of the diagonal morphism R → R2 on the real line is given by the line which is a graph of the equation y=x. The diagonal morphism into the infinite product X∞ may provide an injection into the space of sequences valued in X; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions which the image of the diagonal map will fail to satisfy.
In particular, the category of categories has products, and so one finds the diagonal functor Δ: C → C×C given by Δ(a) = (a,a), the ordered pair for any object a in C. This functor can be employed to give a succinct alternate description of the product of objects within the category C: a product a×b is a universal arrow from Δ to (a,b). The arrow comprises the projection maps.
More generally, in any functor category CJ (here J should be thought of as a small index category), for each object a in C, there is a constant functor Δa which maps each object j in J to a Δa(j) = a and maps each morphism j → k in J to the identity morphism on a. The diagonal functor Δ: C → CJ assigns to each object of C the constant functor at that object (Δ(a) = Δa ∈ CJ), and to each morphism f: a → b in C the obvious natural transformation in CJ (given by ηj = f). In the case that J is a discrete category with two objects, the diagonal functor C → C×C is recovered.
Diagonal functors provide a way to define limits and colimits of functors. The limit of any functor F: J → C is a universal arrow from Δ to F and a colimit is a universal arrow F → Δ. If every functor from J to C has a limit (which will be the case if C is complete), then the operation of taking limits is itself a functor from CJ to C. The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor from the discrete two object category is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.
- δa: a → a×a,
satisfying
- πkδa = ida for k=1,2, where
- πk
is the canonical projection morphism to the k-th component. The existence of this morphism is a consequence of the universal property which characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.
For concrete categories, the diagonal morphism can be simply described by its action on elements x of the object a. Namely, δa(x) = (x,x), the ordered pair formed from x. The reason for the name is that the graph of such a diagonal morphism is diagonal (whenever it makes sense), for example the graph of the diagonal morphism R → R2 on the real line is given by the line which is a graph of the equation y=x. The diagonal morphism into the infinite product X∞ may provide an injection into the space of sequences valued in X; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions which the image of the diagonal map will fail to satisfy.
In particular, the category of categories has products, and so one finds the diagonal functor Δ: C → C×C given by Δ(a) = (a,a), the ordered pair for any object a in C. This functor can be employed to give a succinct alternate description of the product of objects within the category C: a product a×b is a universal arrow from Δ to (a,b). The arrow comprises the projection maps.
More generally, in any functor category CJ (here J should be thought of as a small index category), for each object a in C, there is a constant functor Δa which maps each object j in J to a Δa(j) = a and maps each morphism j → k in J to the identity morphism on a. The diagonal functor Δ: C → CJ assigns to each object of C the constant functor at that object (Δ(a) = Δa ∈ CJ), and to each morphism f: a → b in C the obvious natural transformation in CJ (given by ηj = f). In the case that J is a discrete category with two objects, the diagonal functor C → C×C is recovered.
Diagonal functors provide a way to define limits and colimits of functors. The limit of any functor F: J → C is a universal arrow from Δ to F and a colimit is a universal arrow F → Δ. If every functor from J to C has a limit (which will be the case if C is complete), then the operation of taking limits is itself a functor from CJ to C. The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor from the discrete two object category is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
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In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion.
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In category theory, one defines products to generalize constructions such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.
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projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following:
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- In set theory, an operation typified by the j th projection map, written projj
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universal properties. Universal properties are studied abstractly using the language of category theory.
This article gives a general treatment of universal properties.
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This article gives a general treatment of universal properties.
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In the jargon of mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P.
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In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e.
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In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
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image of f is a monomorphism satisfying the following universal property:
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- There exists a morphism such that f = hg.
- For any object Z with a morphism and a monomorphism such that f = lk
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In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics.
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subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
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In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept.
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relation or relationship is a generalization of 2-place relations, such as the relation of equality, denoted by the sign "=" in a statement like "5 + 7 = 12," or the relation of order,
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domain is most often defined as the set of values, D for which a function is defined.[1] A function that has a domain N is said to be a function over N, where N is an arbitrary set.
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equal if and only if they are precisely the same in every way. The complementary notion is distinctness. This defines a binary relation, equality, denoted by the sign of equality "=" in such a way that the statement "x = y" means that x
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In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions.
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In mathematics, an ordered pair is a collection of two not necessarily distinct objects, one of which is distinguished as the first coordinate (or first entry or
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graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc.
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In mathematics, the real line is simply the set R of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space.
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The word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.
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non-injective function.]] In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y
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In functional analysis and related areas of mathematics, a sequence space is an important class of function space.
The set of all functions from the natural numbers to complex numbers, which can naturally be identified with the set of all possible infinite sequences with
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The set of all functions from the natural numbers to complex numbers, which can naturally be identified with the set of all possible infinite sequences with
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sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence.
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Convergent Series (ISBN 0-7088-8062-2) is a collection of science fiction short stories by Larry Niven, published in 1979. It is also the name of one of the short stories in that collection.
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In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories.
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functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories.
Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological
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Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological
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functor category are natural transformations between functors. Functor categories are of interest for two main reasons:
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- many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable;
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In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion.
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In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well.
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natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors".
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