57 relations: Abstract algebra, Adjoint functors, Algebraic geometry, Associative property, Beck's monadicity theorem, Cambridge University Press, Category of groups, Category of sets, Category theory, Charles Wells (mathematician), Closure operator, Coherence condition, Commutative diagram, Concatenation, Descent (mathematics), Distributive law between monads, Dual (category theory), Equivalence of categories, Equivalence relation, Fiber (mathematics), Forgetful functor, Free group, Functional programming, Functor, Galois connection, Group (mathematics), Identity element, If and only if, Image (mathematics), Interior algebra, Intuitionistic logic, Kleisli category, Lawvere theory, Mathematical model, Michael Barr (mathematician), Modal logic, Monad (category theory), Monad (functional programming), Monoid, Monoid (category theory), Monoidal category, Natural transformation, Opposite category, Partially ordered set, Polyad, Power set, Roger Godement, Saunders Mac Lane, Singleton (mathematics), Strict 2-category, ..., Strong monad, Subcategory, Tensor algebra, Topos, Union (set theory), Universal algebra, Variety (universal algebra). Expand index (7 more) »
Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
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Adjoint functors
In mathematics, specifically category theory, adjunction is a possible relationship between two functors.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Associative property
In mathematics, the associative property is a property of some binary operations.
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Beck's monadicity theorem
In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by in about 1964.
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms.
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Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets.
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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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Charles Wells (mathematician)
Charles Wells (* 4 May 1937 in Atlanta, Georgia; † 17 June 2017) was an American mathematician known for his fundamental contributions to category theory.
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Closure operator
In mathematics, a closure operator on a set S is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y\subseteq S |- | X \subseteq \operatorname(X) | (cl is extensive) |- | X\subseteq Y \Rightarrow \operatorname(X) \subseteq \operatorname(Y) | (cl is increasing) |- | \operatorname(\operatorname(X)).
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Coherence condition
In mathematics, and particularly category theory a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal.
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Commutative diagram
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.
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Concatenation
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end.
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Descent (mathematics)
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology.
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Distributive law between monads
In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one.
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Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop.
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Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Fiber (mathematics)
In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context.
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Forgetful functor
In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output.
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Free group
In mathematics, the free group FS over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st.
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Functional programming
In computer science, functional programming is a programming paradigm—a style of building the structure and elements of computer programs—that treats computation as the evaluation of mathematical functions and avoids changing-state and mutable data.
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Functor
In mathematics, a functor is a map between categories.
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Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets).
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Identity element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
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Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set.
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Intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof.
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Kleisli category
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free ''T''-algebras.
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Lawvere theory
In category theory, a Lawvere theory (named after American mathematician William Lawvere) is a category which can be considered a categorical counterpart of the notion of an equational theory.
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Mathematical model
A mathematical model is a description of a system using mathematical concepts and language.
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Michael Barr (mathematician)
Michael Barr (born January 22, 1937) is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University.
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Modal logic
Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality.
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Monad (category theory)
In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations.
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Monad (functional programming)
In functional programming, a monad is a design pattern that defines how functions, actions, inputs, and outputs can be used together to build generic types, with the following organization.
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Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
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Monoid (category theory)
In category theory, a monoid (or monoid object) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms.
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Monoidal category
In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.
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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.
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Opposite category
In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism.
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Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
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Polyad
In mathematics, polyad is a concept of category theory introduced by Jean Bénabou in generalising monads.
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Power set
In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.
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Roger Godement
Roger Godement (October 1, 1921 – July 21, 2016) was a French mathematician, known for his work in functional analysis as well as his expository books.
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Saunders Mac Lane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.
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Singleton (mathematics)
In mathematics, a singleton, also known as a unit set, is a set with exactly one element.
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Strict 2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category.
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Strong monad
In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation tA,B: A ⊗ TB → T(A ⊗ B), called (tensorial) strength, such that the diagrams commute for every object A, B and C (see Definition 3.2 in). If the monoidal category (C, ⊗, I) is closed then a strong monad is the same thing as a C-enriched monad.
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Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms.
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Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.
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Topos
In mathematics, a topos (plural topoi or, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).
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Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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Universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
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Variety (universal algebra)
In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities.
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Algebra for a monad, Algebra over a monad, Comonad, Cotriple, Eilenberg-Moore algebra, Eilenberg-Moore category, Eilenberg–Moore algebra, Eilenberg–Moore category, Monad (math), Monadic adjunction, Monadic functor, T-algebra, Triple (category theory), Tripleable.
References
[1] https://en.wikipedia.org/wiki/Monad_(category_theory)