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Dualizing Object
In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object \bot. The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality. Definition Let C be a symmetric monoidal closed category. For any object ''A'' and \bot, there exists a morphism :\partial_:A\to(A\Rightarrow\bot)\Rightarrow\bot defined as the image by the bijection defining the monoidal closure :\mathrm((A\Rightarrow\bot)\otimes A,\bot)\cong\mathrm(A,(A\Rightarrow\bot)\Rightarrow\bot) of the morphism :\mathrm_\circ\gamma_ : (A\Rightarrow\bot)\otimes A\to\bot where \gamma is the ''symmetry'' of the tensor product. An object \bot of the category C is called dualizing when the associated morphism \partial_ is an isomorphism for every object ''A'' of the category C. Equivalently, a *-autonomous category is a symmetric monoidal category ''C'' together with a functor (-)^*:C^\to ... [...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] |
Semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa. Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order. A lattice is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biclosed Monoidal Category
In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic example is the category of sets, Set, where the monoidal product of sets A and B is the usual cartesian product A \times B, and the internal Hom B^A is the set of functions from A to B. A non- cartesian example is the category of vector spaces, ''K''-Vect, over a field K. Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another. The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chu Space
Chu spaces generalize the notion of topological space by dropping the requirements that the set of open sets be closed under union and finite intersection, that the open sets be extensional, and that the membership predicate (of points in open sets) be two-valued. The definition of continuous function remains unchanged other than having to be worded carefully to continue to make sense after these generalizations. The name is due to Po-Hsiang Chu, who originally constructed a verification of autonomous categories as a graduate student under the direction of Michael Barr in 1979. Definition Understood statically, a Chu space (''A'', ''r'', ''X'') over a set ''K'' consists of a set ''A'' of points, a set ''X'' of states, and a function ''r'' : ''A'' × ''X'' → ''K''. This makes it an ''A'' × ''X'' matrix with entries drawn from ''K'', or equivalently a ''K''-valued binary relation between ''A'' and ''X'' (ordinary binary relations being 2-valued). Understood dynamically, Chu sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Barr (mathematician)
Michael Barr (born January 22, 1937) is an American mathematician who is the Peter Redpath Emeritus Professor of Pure Mathematics at McGill University. Early life and education He was born in Philadelphia, Pennsylvania, and graduated from the 202nd class of Central High School in June 1954. He graduated from the University of Pennsylvania in February 1959 and received a PhD from the same school in June 1962. Career Barr studied mathematics at the University of Pennsylvania, graduating with a bachelor's degree in 1959 and a doctorate in 1962 under David Kent Harrison ''(Cohomology of Commutative Algebras''). He was then an instructor at Columbia University and from 1964 Assistant Professor and later Associate Professor at the University of Illinois Urbana-Champaign. In 1968 he became Associate Professor and in 1972 Professor at McGill University. In 1967 and 1975/76 he was a visiting scientist at ETH Zurich and in 1970/71 at the University of Fribourg and in 1989/90 a visiting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be Irreducible component, irreducible, which means that it is not the Union (set theory), union of two smaller Set (mathematics), sets that are Closed set, closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-adic Sheaf
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is Reflexive relation, reflexive, antisymmetric relation, antisymmetric, and Transitive relation, transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Yves Girard
Jean-Yves Girard (; born 1947) is a French logician working in proof theory. He is a research director (emeritus) at the mathematical institute of University of Aix-Marseille, at Luminy. Biography Jean-Yves Girard is an alumnus of the École normale supérieure de Saint-Cloud. He made a name for himself in the 1970s with his proof of strong normalization in a system of second-order logic called System F. This result gave a new proof of Takeuti's conjecture, which was proven a few years earlier by William W. Tait, Motō Takahashi and Dag Prawitz. For this purpose, he introduced the notion of "reducibility candidate" ("candidat de réducibilité"). He is also credited with the discovery of Girard's paradox, linear logic, the geometry of interaction, ludics, and (satirically) the mustard watch. He obtained the CNRS Silver Medal in 1983 and is a member of the French Academy of Sciences. Bibliography * * * * Jean-Yves Girard (2011). ''The Blind Spot: Lectures on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Monoidal Category
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sense, naturally isomorphic to B\otimes A for all objects A and B of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field ''k,'' using the ordinary tensor product of vector spaces. Definition A symmetric monoidal category is a monoidal category (''C'', ⊗, ''I'') such that, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism s_: A \otimes B \to B \otimes A called the ''swap map'' that is natural in both ''A'' and ''B'' and such that the following diagrams commute: *The unit coherence: *: *The associativity coherence: *: *The inverse law: *: In the diagrams above, ''a'', ''l'', and ''r'' are the associativity isomorphism, the left unit i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Logic
Linear logic is a substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics, and quantum physics (because linear logic can be seen as the logic of quantum information theory), as well as linguistics, particularly because of its emphasis on resource-boundedness, duality, and interaction. Linear logic lends itself to many different presentations, explanations, and intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and weakening are carefully controlled. Operationally, this means that logical deduction is no longer merely about an ever-expanding collection of pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
