Category theory Category theory In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...more on Wikipedia about "Groupoid"
In mathematics, specifically in category theory, Hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called Hom-functors and have numerous applications in category theory and other branches of mathematics. ...more on Wikipedia about "Hom functor"
Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: ...more on Wikipedia about "Image (category theory)"
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there exists a single morphism X → T. Initial objects are also called coterminal and terminal objects are also called final. If an object is both initial and terminal, we call it a zero object. ...more on Wikipedia about "Initial object"
In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and (dually) cogenerators are objects which envelope other objects as an approximation. When working with unfamiliar algebraic objects, one can use these to approximate with the more familiar. ...more on Wikipedia about "Injective cogenerator"
In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category, but we will initially only consider inverse limits of groups. ...more on Wikipedia about "Inverse limit"
In mathematics, an isomorphism ( Greek:isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e. structure-preserving mappings. ...more on Wikipedia about "Isomorphism"
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In mathematics the Karoubi envelope or Cauchy completion of a category C is a classification of the idempotents of C. It is named for the French mathematician Max Karoubi. ...more on Wikipedia about "Karoubi envelope"
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism f : X → Y is the "most general" morphism k : K → X which, when composed with f, yields zero. ...more on Wikipedia about "Kernel (category theory)"
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...more on Wikipedia about "Limit (category theory)"
This is a list of category theory topics, by Wikipedia page. ...more on Wikipedia about "List of category theory topics"
(List of publications in mathematics) === Euclid's Elements=== ...more on Wikipedia about "List of publications in mathematics"
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category. ...more on Wikipedia about "Localization of a category"
In mathematics, Mitchell's embedding theorem is an important result about abelian categories; it states that these categories, while rather abstractly defined, are all quite concrete categories of modules. This allows one to use element-wise diagram chasing proofs in arbitrary abelian categories. ...more on Wikipedia about "Mitchell's embedding theorem"
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In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows'): ' weak equivalences', ' fibrations' and ' cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes ( derived category theory). This concept was introduced in 1967 by Daniel G. Quillen. ...more on Wikipedia about "Model category"
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. ...more on Wikipedia about "Monoid"
In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...more on Wikipedia about "Monomorphism"
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...more on Wikipedia about "Morphism"
In category theory, a natural number object (nno) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object (alternately, a topos), an nno N is given by: ...more on Wikipedia about "Natural number object"
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 natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so called functor categories. Natural transformations are, after categories and functors, one of the most basic notions of categorical algebra and consequently appear in the majority of its applications. ...more on Wikipedia about "Natural transformation"
In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. ...more on Wikipedia about "Normal morphism"
Pierre Cartier (born in Sedan, France in 1932) is a mathematician - more specifically, a category theorist. ...more on Wikipedia about "Pierre Cartier"
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f(x0) = y0. This is usually denoted ...more on Wikipedia about "Pointed space"
In mathematics, pointless topology (also called point-free or pointfree topology) is an approach to topology which avoids the mentioning of points. ...more on Wikipedia about "Pointless topology"
In mathematics, specifically in category theory, a pre-Abelian category is an additive category that has all kernels and cokernels. ...more on Wikipedia about "Pre-Abelian category"
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