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Anafunctor
An anafunctor is a notion introduced by for ordinary categories that is a generalization of functors. In category theory, some statements require the axiom of choice, but the axiom of choice can sometimes be avoided when using an anafunctor. For example, the statement "every fully faithful and essentially surjective functor is an equivalence of categories" is equivalent to the axiom of choice, but we can usually follow the same statement without the axiom of choice by using anafunctor instead of functor. Definition Span formulation of anafunctors Let and be categories. An anafunctor with domain ( source) and codomain (target) , and between categories and is a category , F, , in a notation F:X \xrightarrow A, is given by the following conditions: *F_0 is surjective on objects. *Let pair F_0:, F, \rightarrow X and F_1:, F, \rightarrow A be functors, a span of ordinary functors (X \leftarrow , F, \rightarrow A), where F_0 is fully faithful. Set-theoretic definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anafunctor (span)
An anafunctor is a notion introduced by for ordinary categories that is a generalization of functors. In category theory, some statements require the axiom of choice, but the axiom of choice can sometimes be avoided when using an anafunctor. For example, the statement "every fully faithful and essentially surjective functor is an equivalence of categories" is equivalent to the axiom of choice, but we can usually follow the same statement without the axiom of choice by using anafunctor instead of functor. Definition Span formulation of anafunctors Let and be categories. An anafunctor with domain ( source) and codomain ( target) , and between categories and is a category , F, , in a notation F:X \xrightarrow A, is given by the following conditions: *F_0 is surjective on objects. *Let pair F_0:, F, \rightarrow X and F_1:, F, \rightarrow A be functors, a span of ordinary functors (X \leftarrow , F, \rightarrow A), where F_0 is fully faithful. Set-theoretic defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anaphase
Anaphase () is the stage of mitosis after the process of metaphase, when replicated chromosomes are split and the newly-copied chromosomes (daughter chromatids) are moved to opposite poles of the cell. Chromosomes also reach their overall maximum condensation in late anaphase, to help chromosome segregation and the re-formation of the nucleus. Anaphase starts when the anaphase promoting complex marks an inhibitory chaperone called securin for destruction by ubiquinylating it. Securin is a protein which inhibits a protease known as separase. The destruction of securin unleashes separase which then breaks down cohesin, a protein responsible for holding sister chromatids together. At this point, three subclasses of microtubule unique to mitosis are involved in creating the forces necessary to separate the chromatids: kinetochore microtubules, interpolar microtubules, and astral microtubules. The centromeres are split, and the sister chromatids are pulled toward the poles by ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Span (category Theory)
In category theory, a span, roof or correspondence is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as morphisms in a category of fractions. The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967). Formal definition A span is a diagram of type \Lambda = (-1 \leftarrow 0 \rightarrow +1), i.e., a diagram of the form Y \leftarrow X \rightarrow Z. That is, let Λ be the category (-1 ← 0 → +1). Then a span in a category ''C'' is a functor ''S'' : Λ → ''C''. This means that a span consists of three objects ''X'', ''Y'' and ''Z'' of ''C'' and morphisms ''f'' : ''X'' → ''Y'' and ''g'' : ''X'' → ''Z'': it is two maps with common ''domain''. The colimit of a span is a pushout. Examples * If ''R'' is a relation between sets ''X'' and ''Y'' (i.e. a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Profunctor
In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. Definition A profunctor (also named distributor by the French school and module by the Sydney school) \,\phi from a category C to a category D, written :\phi \colon C\nrightarrow D, is defined to be a functor :\phi \colon D^\times C\to\mathbf where D^\mathrm denotes the opposite category of D and \mathbf denotes the category of sets. Given morphisms f\colon d\to d', g\colon c\to c' respectively in D, C and an element x\in\phi(d',c), we write xf\in \phi(d,c), gx\in\phi(d',c') to denote the actions. Using the cartesian closure of \mathbf, the category of small categories, the profunctor \phi can be seen as a functor :\hat \colon C\to\hat where \hat denotes the category \mathrm^ of presheaves over D. A correspondence from C to D is a profunctor D\nrightarrow C. Profunctors as categories An equivalent definition of a profunctor \phi \colon C\nright ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set. Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets. In the past, "0" was occasionally used as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inhabited Set
In constructive mathematics, a set A is inhabited if there exists an element a \in A. In classical mathematics, this is the same as the set being nonempty; however, this equivalence is not valid in intuitionistic logic (or constructive logic). Comparison with nonempty sets In classical mathematics, a set is inhabited if and only if it is not the empty set. These definitions diverge in constructive mathematics, however. A set A is if \forall z (z \not \in A) while A is if it is not empty, that is, if \lnot forall z (z \not \in A) It is if \exists z (z \in A). Every inhabited set is a nonempty set (because if a \in A is an inhabitant of A then a \not\in A is false and consequently so is \forall z (z \not \in A)). In intuitionistic logic, the negation of a universal quantifier is weaker than an existential quantifier, not equivalent to it as in classical logic so a nonempty set is not automatically guaranteed to be inhabited. Example Because inhabited sets are the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...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 functor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prophase
Prophase () is the first stage of cell division in both mitosis and meiosis. Beginning after interphase, DNA has already been replicated when the cell enters prophase. The main occurrences in prophase are the condensation of the chromatin reticulum and the disappearance of the nucleolus. Staining and microscopy Microscopy can be used to visualize condensed chromosomes as they move through meiosis and mitosis. Various DNA stains are used to treat cells such that condensing chromosomes can be visualized as the move through prophase. The giemsa G-banding technique is commonly used to identify mammalian chromosomes, but utilizing the technology on plant cells was originally difficult due to the high degree of chromosome compaction in plant cells. G-banding was fully realized for plant chromosomes in 1990. During both meiotic and mitotic prophase, giemsa staining can be applied to cells to elicit G-banding in chromosomes. Silver staining, a more modern technology, in conjunction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Object Of A Category
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 (other)#Mathematics, completion, and duality (mathematics), duality. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
