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List Object
In category theory, an abstract branch of mathematics, and in its applications to logic and theoretical computer science, a list object is an abstract definition of a list, that is, a finite ordered sequence. Formal definition Let C be a category with finite products and a terminal object 1. A list object over an object of C is: # an object , # a morphism : 1 → , and # a morphism : × → such that for any object of with maps : 1 → and : × → , there exists a unique : → such that the following diagram commutes: where〈id, 〉denotes the arrow induced by the universal property of the product when applied to id (the identity on ) and . The notation * ( à la Kleene star) is sometimes used to denote lists over . Equivalent definitions In a category with a terminal object 1, binary coproducts (denoted by +), and binary products (denoted by ×), a list object over can be defined as the initial algebra of the endofunctor that acts on obj ... [...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] |
Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products. Definition Let C be a category and let X_1 and X_2 be objects of C. An object is called the coproduct of X_1 and X_2, written X_1 \sqcup X_2, or X_1 \oplus X_2, or sometimes simply X_1 + X_2, if there exist morphisms i_1 : X_1 \to X_1 \sqcup X_2 and i_2 : X_2 \to X_1 \sqcup X_2 satisfying the following universal property: for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Object Length
A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List The SC Germania List is a German rugby union club from the district List of Hanover, currently playing in the Rugby-Bundesliga. Apart from rugby, the club also offers other sports like tennis, gymnastics and handball. The club has three German ..., German rugby union club Other uses * Angle of list, the leaning to either port or starboard of a ship * List (information), an ordered collection of pieces of information ** List (abstract data type), a method to organize data in computer science * List on Sylt, previously called List, the northernmost village in Germany, on the island of Sylt * ''List'', an alternative term for ''roll'' in flight dynamics * To ''list'' a building, etc., in the UK it means to designate it a listed building that may n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number Object
In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO ''N'' is given by: # a global element ''z'' : 1 → ''N'', and # an arrow ''s'' : ''N'' → ''N'', such that for any object ''A'' of E, global element ''q'' : 1 → ''A'', and arrow ''f'' : ''A'' → ''A'', there exists a unique arrow ''u'' : ''N'' → ''A'' such that: # ''u'' ∘ ''z'' = ''q'', and # ''u'' ∘ ''s'' = ''f'' ∘ ''u''. In other words, the triangle and square in the following diagram commute. The pair (''q'', ''f'') is sometimes called the ''recursion data'' for ''u'', given in the form of a recursive definition: # ⊢ ''u'' (''z'') = ''q'' # ''y'' ∈E ''N'' ⊢ ''u'' (''s'' ''y'') = ''f'' (''u'' (''y'')) The above definition is the universal property of NNOs, meaning they are defined up to canonical isomorphism. If the arrow ''u'' as defined above merely has to exist, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haskell (programming Language)
Haskell () is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research and industrial applications, Haskell has pioneered a number of programming language features such as type classes, which enable type-safe operator overloading, and monadic IO. Haskell's main implementation is the Glasgow Haskell Compiler (GHC). It is named after logician Haskell Curry. Haskell's semantics are historically based on those of the Miranda programming language, which served to focus the efforts of the initial Haskell working group. The last formal specification of the language was made in July 2010, while the development of GHC continues to expand Haskell via language extensions. Haskell is used in academia and industry. , Haskell was the 28th most popular programming language by Google searches for tutorials, and made up less than 1% of active users on the GitHub source code repository. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Programming
In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declarative programming paradigm in which function definitions are Tree (data structure), trees of Expression (computer science), expressions that map Value (computer science), values to other values, rather than a sequence of Imperative programming, imperative Statement (computer science), statements which update the State (computer science), running state of the program. In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local Identifier (computer languages), identifiers), passed as Parameter (computer programming), arguments, and Return value, returned from other functions, just as any other data type can. This allows programs to be written in a Declarative programming, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong And Weak Typing
In computer programming, one of the many ways that programming languages are colloquially classified is whether the language's type system makes it strongly typed or weakly typed (loosely typed). However, there is no precise technical definition of what the terms mean and different authors disagree about the implied meaning of the terms and the relative rankings of the "strength" of the type systems of mainstream programming languages. For this reason, writers who wish to write unambiguously about type systems often eschew the terms "strong typing" and "weak typing" in favor of specific expressions such as "type safety". Generally, a strongly typed language has stricter typing rules at compile time, which implies that errors and exceptions are more likely to happen during compilation. Most of these rules affect variable assignment, function return values, procedure arguments and function calling. Dynamically typed languages (where type checking happens at run time) can also be stro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions. History Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic. Russell's paradox, which was discovered by Bertrand Russell, existed because a set could be defined using "all possible sets", which included itself. Between 1902 and 1908, Bertrand Russell proposed various "theories of type" to fix the problem. By 1908 Russell arrived at a "ramified" theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Constructions
In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants. Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity). General traits The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from terms to terms, as well as terms to types, types to types, and types to terms. The CoC is strongly normalizing, and hence consistent. Usage The CoC has been developed alongsi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Element (mathematics)
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets of . Sets can themselves be elements. For example, consider the set B = \. The elements of are ''not'' 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set \. The elements of a set can be anything. For example, C = \ is the set whose elements are the colors , and . Notation and terminology The relation "is an element of", also called set membership, is denoted by the symbol "∈". Writing :x \in A means that "''x'' is an element of ''A''". Equivalent expressions are "''x'' is a member of ''A''", "''x'' belongs to ''A''", "''x'' is in ''A''" and "''x'' lies in ''A''". The expressions "''A'' includes ''x''" and "''A'' contains ''x''" are also used to mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of morphisms is the composition of functions. Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind. Properties of the category of sets The axioms of a category are satisfied by Set because composition of functions is associative, and because every set ''X'' has an identity function id''X'' : ''X'' → ''X'' which serves as identity element for function composition. The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions as morphisms. Every s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
