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Monoidal Functor
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two ''coherence maps''—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors * The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible. * The coherence maps of strong monoidal functors are invertible. * The coherence maps of strict monoidal functors are identity maps. Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors. Defi ... [...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] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (''n'' + 1)-dimensional manifold ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Adjunction
Suppose that (\mathcal C,\otimes,I) and (\mathcal D,\bullet,J) are two monoidal categories. A monoidal adjunction between two lax monoidal functors :(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) and (G,n):(\mathcal D,\bullet,J)\to(\mathcal C,\otimes,I) is an adjunction (F,G,\eta,\varepsilon) between the underlying functors, such that the natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...s :\eta:1_\Rightarrow G\circ F and \varepsilon:F\circ G\Rightarrow 1_{\mathcal D} are monoidal natural transformations. Lifting adjunctions to monoidal adjunctions Suppose that :(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J) is a lax monoidal functor such that the underlying functor F:\mathcal C\to\mathcal D has a right adjoint G:\mathcal D\to\mathcal C. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Natural Transformation
Suppose that (\mathcal C,\otimes,I) and (\mathcal D,\bullet, J) are two monoidal categories and :(F,m):(\mathcal C,\otimes,I)\to(\mathcal D,\bullet, J) and (G,n):(\mathcal C,\otimes,I)\to(\mathcal D,\bullet, J) are two lax monoidal functors between those categories. A monoidal natural transformation :\theta:(F,m) \to (G,n) between those functors is a natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ... \theta:F \to G between the underlying functors such that the diagrams : and commute for every objects A and B of \mathcal C (see Definition 11 in ). A symmetric monoidal natural transformation is a monoidal natural transformation between symmetric monoidal functors. References {{DEFAULTSORT:Monoidal Natural Transformation Monoidal categories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid Object
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''η'': ''I'' → ''M'' called ''unit'', such that the pentagon diagram : and the unitor diagram : commute. In the above notation, is the identity morphism of , is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop. Suppose that the monoidal category C has a symmetry ''γ''. A monoid ''M'' in C is commutative when . Examples * A monoid object in Set, the category of sets (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense. * A monoid object in Top, the category of topological spaces (with the monoidal structure induced by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applicative Form Of Monoidal Coherence Map 01
Applicative can refer to: *Applicative programming language *Applicative voice The applicative voice (; abbreviated or ) is a grammatical voice that promotes an oblique argument of a verb to the core object argument. It is generally considered a valency-increasing morpheme. The Applicative is often found in agglutinative ... * Applicative functor {{Disambiguation ... [...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] |
Closed Monoidal Categories
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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Quantum Field Theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory. In condensed matter physics, topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states. Overview In a topological field theory, correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of sp ... [...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] |
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