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End (category Theory)
In category theory, an end of a functor S:\mathbf^\times\mathbf\to \mathbf is a universal extranatural transformation from an object ''e'' of X to ''S''. More explicitly, this is a pair (e,\omega), where ''e'' is an object of X and \omega:e\ddot\to S is an extranatural transformation such that for every extranatural transformation \beta : x\ddot\to S there exists a unique morphism h:x\to e of X with \beta_a=\omega_a\circ h for every object ''a'' of C. By abuse of language the object ''e'' is often called the ''end'' of the functor ''S'' (forgetting \omega) and is written :e=\int_c^ S(c,c)\text\int_\mathbf^ S. Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram :\int_c S(c, c) \to \prod_ S(c, c) \rightrightarrows \prod_ S(c, c'), where the first morphism being equalized is induced by S(c, c) \to S(c, c') and the second is induced by S(c', c') \to S(c, c'). Coend The definition of the coend of a functor S:\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extranatural Transformation
(dually co-wedges and co-ends), by setting F (dually G) constant. Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case. See also * Dinatural transformation External links * {{nlab, id=extranatural+transformation References Higher category theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Category
In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of ''all'' limits (even when ''J'' is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists (i.e. limits of diagrams indexed by a finite category ''J''). Dually, a category is finitely cocomplete if all finite colimits exist. Theorems It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (of all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equalizer (mathematics)
In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Definitions Let ''X'' and ''Y'' be sets. Let ''f'' and ''g'' be functions, both from ''X'' to ''Y''. Then the ''equaliser'' of ''f'' and ''g'' is the set of elements ''x'' of ''X'' such that ''f''(''x'') equals ''g''(''x'') in ''Y''. Symbolically: : \operatorname(f, g) := \. The equaliser may be denoted Eq(''f'', ''g'') or a variation on that theme (such as with lowercase letters "eq"). In informal contexts, the notation is common. The definition above used two functions ''f'' and ''g'', but there is no need to restrict to only two functions, or even to only finitely many functions. In general, if F is a set of functions from ''X'' to ''Y'', then the ''equaliser'' of the members of F is the set of elements ''x'' of ''X'' such that, given any two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
