|
Join (category Theory)
In category theory in mathematics, the join of categories is an operation making the category of small categories into a monoidal category. In particular, it takes two small categories to construct another small category. Under the nerve construction, it corresponds to the join of simplicial sets. Definition For small categories \mathcal and \mathcal, their ''join'' \mathcal\star\mathcal is the small category with:Joyal 2008, p. 241 : \operatorname(\mathcal\star\mathcal) =\operatorname(\mathcal)\sqcup\operatorname(\mathcal); : \operatorname_(X,Y) :=\begin \operatorname_(X,Y); & X,Y\in\operatorname(\mathcal) \\ \operatorname_(X,Y); & X,Y\in\operatorname(\mathcal) \\ \; & X\in\operatorname(\mathcal), Y\in\operatorname(\mathcal) \\ \emptyset; & X\in\operatorname(\mathcal), Y\in\operatorname(\mathcal) \end. The join defines a functor -\star-\colon \mathbf\times\mathbf\rightarrow \mathbf, which together with the empty category as unit element makes the category of small categorie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. 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. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operation (mathematics)
In mathematics, an operation is a function from a set to itself. For example, an operation on real numbers will take in real numbers and return a real number. An operation can take zero or more input values (also called "'' operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations. A partial operation is defined similarly to an operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Small Categories
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. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial object of Cat is the ''empty category'' 0, which is the category of no objects and no morphisms. The terminal object is the ''terminal category'' or ''trivial category'' 1 with a single object and morphism. at nLab The category Cat is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (category theory), object ''I'' that is both a left identity, left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagram (category theory), diagrams commutative diagram, commute. The ordinary tensor product makes vector spaces, abelian groups, module (mathematics), ''R''-modules, or algebra (ring theory), ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small category, small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nerve (category Theory)
In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a topological space, called the classifying space of the category ''C''. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory. Motivation The nerve of a category is often used to construct topological versions of moduli spaces. If ''X'' is an object of ''C'', its moduli space should somehow encode all objects isomorphic to ''X'' and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Join (simplicial Sets)
In higher category theory in mathematics, the join of simplicial sets is an operation making the category of simplicial sets into a monoidal category. In particular, it takes two simplicial sets to construct another simplicial set. It is closely related to the diamond operation and used in the construction of the twisted diagonal. Under the nerve construction, it corresponds to the join of categories and under the geometric realization, it corresponds to the join of topological spaces. Definition For natural numbers m,p,q\in\mathbb, one has the identity:Cisinski 2019, 3.4.12. : \operatorname( +q+1 =\prod_\operatorname( \times\operatorname( , which can be extended by colimits to a functor a functor -*-\colon \mathbf\times\mathbf\rightarrow \mathbf, which together with the empty simplicial set as unit element makes the category of simplicial sets \mathbf into a monoidal category. For simplicial set X and Y, their ''join'' X*Y is the simplicial set: : (X*Y)_n =\prod_X_ ... [...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 Mathematical object, object X in ''C'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Category (category Theory)
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. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial object of Cat is the ''empty category'' 0, which is the category of no objects and no morphisms. The terminal object is the ''terminal category'' or ''trivial category'' 1 with a single object and morphism. at nLab The category Cat is itself a , and therefore not an object of itself. In order to avoid problems analogous to |
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 c in \mathcal and d in \mathcal, a bijection between the respective morphism sets :\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
