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Presheaf (category Theory)
In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on C into a category, and is an example of a functor category. It is often written as \widehat = \mathbf^. A functor into \widehat is sometimes called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, ''A'') for some object ''A'' of C is called a representable presheaf. Some authors refer to a functor F\colon C^\mathrm\to\mathbf as a \mathbf-valued presheaf. Examples * A simplicial set is a Set-valued presheaf on the simplex category C=\Delta. Properties * When C is a small category, the functor category \widehat=\mathbf^ is cartesian clos ... [...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 com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex Category
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition The simplex category is usually denoted by \Delta. There are several equivalent descriptions of this category. \Delta can be described as the category of ''non-empty finite ordinals'' as objects, thought of as totally ordered sets, and ''(non-strictly) order-preserving functions'' as morphisms. The objects are commonly denoted = \ (so that is the ordinal n+1 ). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on \Delta, that is a contravariant functor from \Delta to another category. For instance, simplicial sets are contravariant with the codomain category being the catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Theorem (category Theory)
In category theory, a branch of mathematics, the density theorem states that every presheaf of sets is a colimit of representable presheaves in a canonical way. For example, by definition, a simplicial set is a presheaf on the simplex category Δ and a representable simplicial set is exactly of the form \Delta^n = \operatorname(-, (called the standard ''n''-simplex) so the theorem says: for each simplicial set ''X'', :X \simeq \varinjlim \Delta^n where the colim runs over an index category determined by ''X''. Statement Let ''F'' be a presheaf on a category ''C''; i.e., an object of the functor category \widehat = \mathbf(C^\text, \mathbf). For an index category over which a colimit will run, let ''I'' be the category of elements of ''F'': it is the category where # an object is a pair (U, x) consisting of an object ''U'' in ''C'' and an element x \in F(U), # a morphism (U, x) \to (V, y) consists of a morphism u: U \to V in ''C'' such that (Fu)(y) = x. It comes with the forg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit And Colimit Of Presheaves
In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category ''C'' is a limit or colimit in the functor category \widehat = \mathbf(C^, \mathbf). The category \widehat admits small limits and small colimits. Explicitly, if f: I \to \widehat is a functor from a small category ''I'' and ''U'' is an object in ''C'', then \varinjlim_ f(i) is computed pointwise: :(\varinjlim f(i))(U) = \varinjlim f(i)(U). The same is true for small limits. Concretely this means that, for example, a fiber product exists and is computed pointwise. When ''C'' is small, by the Yoneda lemma In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (vie ..., one can view ''C'' as the full subcategory of \widehat. If \eta: C \to D is a functor, if f: I \to C is a functor from a small catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colimit
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 category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (category Theory)
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 fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hom Functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Formal definition Let ''C'' be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes). For all objects ''A'' and ''B'' in ''C'' we define two functors to the category of sets as follows: : The functor Hom(–, ''B'') is also called the '' functor of points'' of the object ''B''. Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms. The pair of functors Hom(''A'', –) and Hom(–, ''B'') are related in a natural manner. For any pair of morphisms ''f'' : ''B'' → ''B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda Embedding
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Generalities The Yoneda lemma suggests that instead of studying the locally small category \mathcal , one should study the category of all functors of \mathcal into \mathbf (the category of se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Full And Faithful Functors
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor. Formal definitions Explicitly, let ''C'' and ''D'' be ( locally small) categories and let ''F'' : ''C'' → ''D'' be a functor from ''C'' to ''D''. The functor ''F'' induces a function :F_\colon\mathrm_(X,Y)\rightarrow\mathrm_(F(X),F(Y)) for every pair of objects ''X'' and ''Y'' in ''C''. The functor ''F'' is said to be *faithful if ''F''''X'',''Y'' is injectiveJacobson (2009), p. 22 *full if ''F''''X'',''Y'' is surjectiveMac Lane (1971), p. 14 *fully faithful (= full and faithful) if ''F''''X'',''Y'' is bijective for each ''X'' and ''Y'' in ''C''. A mnemonic for remembering the term "full" is that the image of the function fills the codomain; a mnemonic for remembering the term "faithful" is that you can trust (have faith) that F(X)=F(Y) implies X=Y. Properties A faithful fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Quantifier
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by the turned A (∀) logical operator symbol, which, when used together with a predicate variable, is called a universal quantifier ("", "", or sometimes by "" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on quantification (logic). The universal quantifier is encoded as in Unicode, and as \forall in LaTeX and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Adjoint
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic. As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be the supremum of the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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