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- Category theory

K-theory of a category

In algebraic K-theory, the K-theory of a category C (usually equipped with some kind of additional data) is a sequence of abelian groups Ki(C) associated to it. If C is an abelian category, there is n

Lifting property

In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties

Seifert–Van Kampen theorem

In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fun

Pseudo-abelian category

In mathematics, specifically in category theory, a pseudo-abelian category is a category that is preadditive and is such that every idempotent has a kernel. Recall that an idempotent morphism is an en

Setoid

In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied especially in proof

Overcategory

In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They we

Cosmos (category theory)

In the area of mathematics known as category theory, a cosmos is a symmetric closed monoidal category that is complete and cocomplete. Enriched category theory is often considered over a cosmos.

Exact completion

In category theory, a branch of mathematics, the exact completion constructs a Barr-exact category from any finitely complete category. It is used to form the effective topos and other .

Monad (category theory)

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor m

Posetal category

In mathematics, specifically category theory, a posetal category, or thin category, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered c

Lax natural transformation

In the mathematical field of category theory, specifically the theory of 2-categories, a lax natural transformation is a kind of morphism between 2-functors.

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 simplic

Element (category theory)

In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of def

Commutative diagram

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. It is said

Cosheaf

In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimits is a functor F from the category of open subsets of a topological space X (more precisely its nerve)

Filtered category

In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered c

Factorization system

In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situa

Glossary of category theory

This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.)
* Notes on foundations: In many expositions (e.g., Vistoli), the set-theoretic

Induced homomorphism

In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a t

Kernel (category theory)

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kern

Gamma-object

In mathematics, a Γ-object of a pointed category C is a contravariant functor from Γ to C. The basic example is Segal's so-called Γ-space, which may be thought of as a generalization of simplicial abe

Timeline of category theory and related mathematics

This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as:
* Categories of abstract algebraic structures including representation theory and univers

Equivalence of categories

In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are num

Groupoid object

In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the mu

Karoubi envelope

In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi e

Size functor

Given a size pair where is a manifold of dimension and is an arbitrary real continuous function definedon it, the -th size functor, with , denoted by , is the functor in , where is the category of ord

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

Structural Ramsey theory

In mathematics, structural Ramsey theory is a categorical generalisation of Ramsey theory, rooted in the idea that many important results of Ramsey theory have "similar" logical structure. The key obs

Center (category theory)

In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the notion of the center of a monoid, group, or ring

Subcategory

In mathematics, specifically category theory, 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 compositi

Internal category

In mathematics, more specifically in category theory, internal categories are a generalisation of the notion of small category, and are defined with respect to a fixed . If the ambient category is tak

Higher-dimensional algebra

In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract

Krull–Schmidt category

In category theory, a branch of mathematics, a Krull–Schmidt category is a generalization of categories in which the Krull–Schmidt theorem holds. They arise, for example, in the study of finite-dimens

Diagonal functor

In category theory, a branch of mathematics, the diagonal functor is given by , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the pr

Lift (mathematics)

In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such that f = g∘h. We say that f factors through h. A

Grothendieck category

In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for

Category algebra

In category theory, a field of mathematics, a category algebra is an associative algebra, defined for any locally finite category and commutative ring with unity. Category algebras generalize the noti

Image (category theory)

In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

Descent (mathematics)

In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with so

Codensity monad

In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad to a wide class of functors.

Initial algebra

In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework for induction and recursion.

Tower of objects

In category theory, a branch of abstract mathematics, a tower is defined as follows. Let be the poset of whole numbers in reverse order, regarded as a category. A (countable) tower of objects in a cat

Indiscrete category

An indiscrete category is a category C in which every hom-set C(X, Y) is a singleton. Every class X gives rise to an indiscrete category whose objects are the elements of X such that for any two objec

Extensive category

In mathematics, an extensive category is a category C with finite coproducts that are disjoint and well-behaved with respect to pullbacks. Equivalently, C is extensive if the coproduct functor from th

Tame abstract elementary class

In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tamene

Polyad

In mathematics, polyad is a concept of category theory introduced by Jean Bénabou in generalising monads. A polyad in a bicategory D is a bicategory morphism Φ from a locally punctual bicategory C to

Day convolution

In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in

Simplicial localization

In category theory, a branch of mathematics, the simplicial localization of a category C with respect to a class W of morphisms of C is a simplicial category LC whose is the localization of C with res

Stack (mathematics)

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to

Waldhausen category

In mathematics, a Waldhausen category is a category C equipped with some additional data, which makes it possible to construct the K-theory spectrum of C using a so-called S-construction. It's named a

Bousfield localization

In category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak

Krohn–Rhodes theory

In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of eleme

Grothendieck construction

The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory.

Categories for the Working Mathematician

Categories for the Working Mathematician (CWM) is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was fi

Fusion category

In mathematics, a fusion category is a category that is rigid, semisimple, -linear, monoidal and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If

Pseudoalgebra

In algebra, given a 2-monad T in a 2-category, a pseudoalgebra for T is a 2-category-version of algebra for T, that satisfies the laws up to coherent isomorphisms.

Skeleton (category theory)

In mathematics, a skeleton of a category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equival

Symplectic category

In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into

List of types of functions

Functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function.

Isomorphism-closed subcategory

In category theory, a branch of mathematics, a subcategory of a category is said to be isomorphism closed or replete if every -isomorphism with belongs to This implies that both and belong to as well.

Segal category

In mathematics, a Segal category is a model of an infinity category introduced by , based on work of Graeme Segal in 1974.

Coimage

In algebra, the coimage of a homomorphism is the quotient of the domain by the kernel. The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies. M

Corecursion

In computer science, corecursion is a type of operation that is dual to recursion. Whereas recursion works analytically, starting on data further from a base case and breaking it down into smaller dat

Fibred category

Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in

Conservative functor

In category theory, a branch of mathematics, a conservative functor is a functor such that for any morphism f in C, F(f) being an isomorphism implies that f is an isomorphism.

Quantaloid

In mathematics, a quantaloid is a category enriched over the category Sup of suplattices. In other words, for any objects a and b the morphism object between them is not just a set but a complete latt

Cokernel

The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f. Cokernels a

Induced homomorphism (quotient group)

No description available.

Beck's monadicity theorem

In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by Jonathan Mock Beck in about 1964. It is often stated in dual

Product category

In the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. P

T-structure

In the branch of mathematics called homological algebra, a t-structure is a way to axiomatize the properties of an abelian subcategory of a derived category. A t-structure on consists of two subcatego

Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object is said to be embedde

Refinement (category theory)

In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex

Freyd cover

In the mathematical discipline of category theory, the Freyd cover or scone category is a construction that yields a set-like construction out of a given category. The only requirement is that the ori

Bundle (mathematics)

In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology.

Categorical quantum mechanics

Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects

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 propert

Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see below). This functor makes it possible to thin

Burnside category

In category theory and homotopy theory the Burnside category of a finite group G is a category whose objects are finite G-sets and whose morphisms are (equivalence classes of) spans of G-equivariant m

AB5 category

In mathematics, Alexander Grothendieck in his "Tôhoku paper" introduced a sequence of axioms of various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian cat

Grothendieck's Galois theory

In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the

Accessible category

The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to

Esquisse d'un Programme

"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the

F-coalgebra

In mathematics, specifically in category theory, an -coalgebra is a structure defined according to a functor , with specific properties as defined below. For both algebras and coalgebras, a functor is

Categorical set theory

Categorical set theory is any one of several versions of set theory developed from or treated in the context of mathematical category theory.

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 mor

Hylomorphism (computer science)

In computer science, and in particular functional programming, a hylomorphism is a recursive function, corresponding to the composition of an anamorphism (which first builds a set of results; also kno

Indexed category

In category theory, a branch of mathematics, a C-indexed category is a pseudofunctor from Cop to Cat, where Cat is a 2-category of categories. Any indexed category has an associated Grothendieck const

Model category

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfyin

Enriched category

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observa

R-algebroid

In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie al

Homotopy colimit and limit

In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category . The main idea is this: if we

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

Catamorphism

In category theory, the concept of catamorphism (from the Ancient Greek: κατά "downwards" and μορφή "form, shape") denotes the unique homomorphism from an initial algebra into some other algebra. In f

Groupoid

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can b

Coinduction

In computer science, coinduction is a technique for defining and proving properties of systems of concurrent interacting objects. Coinduction is the mathematical dual to structural induction. Coinduct

Cotriple homology

In algebra, given a category C with a cotriple, the n-th cotriple homology of an object X in C with coefficients in a functor E is the n-th homotopy group of the E of the augmented simplicial object i

Duality theory for distributive lattices

In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pair

Gabriel–Popescu theorem

In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu. It characterizes certain abelian categories (the G

Giraud subcategory

In mathematics, Giraud subcategories form an important class of subcategories of Grothendieck categories. They are named after Jean Giraud.

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. Typically the elementary mo

Section (category theory)

In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism.In other words, if and are morphisms whose compositi

Well-pointed category

In category theory, a category with a terminal object is well-pointed if for every pair of arrows such that , there is an arrow such that . (The arrows are called the global elements or points of the

Localizing subcategory

In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to

Sketch (mathematics)

In the mathematical theory of categories, a sketch is a category D, together with a set of cones intended to be limits and a set of cocones intended to be colimits. A model of the sketch in a category

Semiautomaton

In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, a

Mac Lane coherence theorem

In category theory, a branch of mathematics, Mac Lane coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. More precisely (cf. ), it states every commutes, where "for

Sieve (category theory)

In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. I

Localization of a category

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 th

Cone (category theory)

In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.

Lax functor

In category theory, a discipline within mathematics, the notion of lax functor between bicategories generalizes that of functors between categories. Let C,D be bicategories. We denote composition in d

Kan extension

Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan,

Subquotient

In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theor

Injective object

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy

Limit and colimit of presheaves

In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category C is a limit or colimit in the functor category . The category admits small limits and small colimits. Exp

Fraïssé limit

In mathematical logic, specifically in the discipline of model theory, the Fraïssé limit (also called the Fraïssé construction or Fraïssé amalgamation) is a method used to construct (infinite) mathema

Dialectica space

Dialectica spaces are a categorical way of constructing models of linear logic. They were introduced by Valeria de Paiva, Martin Hyland's student, in her doctoral thesis, as a way of modeling both lin

Graded category

If is a category, then a -graded category is a category together with a functor. Monoids and groups can be thought of as categories with a single object. A monoid-graded or group-graded category is th

Allegory (mathematics)

In the mathematical field of category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations between them. Allegories can be used as an abstr

Stable module category

In representation theory, the stable module category is a category in which projectives are "factored out."

Anamorphism

In computer programming, an anamorphism is a function that generates a sequence by repeated application of the function to its previous result. You begin with some value A and apply a function f to it

Cotangent complex

In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If is a mor

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. Given a statement regarding the ca

Categorification

In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, function

Endomorphism ring

In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endom

Quotient category

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous

Multicategory

In mathematics (especially category theory), a multicategory is a generalization of the concept of category that allows morphisms of multiple arity. If morphisms in a category are viewed as analogous

Projective cover

In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of in

Pointless topology

In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points, and in which the lattices of open sets are

Generator (category theory)

In mathematics, specifically category theory, a family of generators (or family of separators) of a category is a collection of objects in , such that for any two distinct morphisms in , that is with

Essential monomorphism

In mathematics, specifically category theory, an essential monomorphism is a monomorphism f in a category C such that for a morphism g in C, the morphism is a monomorphism only when g is a monomorphis

Grothendieck's relative point of view

Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on

Segal space

In mathematics, a Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered as a simp

Spherical category

In category theory, a branch of mathematics, a spherical category is a pivotal category (a monoidal category with traces) in which left and right traces coincide.Spherical fusion categories give rise

Abstract nonsense

In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theo

Operad

In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose

Finitely generated object

In category theory, a finitely generated object is the quotient of a free object over a finite set, in the sense that it is the target of a regular epimorphism from a free object that is free on a fin

Monoid

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid

Corestriction

In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a

Category of representations

In representation theory, the category of representations of some algebraic structure A has the representations of A as objects and equivariant maps as morphisms between them. One of the basic thrusts

Quiver (mathematics)

In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation V

H-object

In mathematics, specifically homotopical algebra, an H-object is a categorical generalization of an H-space, which can be defined in any category with a product and an initial object . These are usefu

Brown's representability theorem

In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW comple

Weak factorization system

No description available.

Inserter category

In category theory, a branch of mathematics, the inserter category is a variation of the comma category where the two functors are required to have the same domain category.

Abstract elementary class

In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure

Applied category theory

Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer science, physics (in particular quantum mec

Isbell conjugacy

Isbell conjugacy (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986.

DisCoCat

No description available.

Semigroupoid

In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small category, except possibly for the requirement that t

Pointed set

In mathematics, a pointed set (also based set or rooted set) is an ordered pair where is a set and is an element of called the base point, also spelled basepoint. Maps between pointed sets and – calle

Hopfian object

In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism. The dual notion is that of a cohopfian object

Outline of category theory

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts,

Polygraph (mathematics)

In mathematics, and particularly in category theory, a polygraph is a generalisation of a directed graph. It is also known as a computad. They were introduced as "polygraphs" by and as "computads" by

Stable model category

In category theory, a branch of mathematics, a stable model category is a pointed model category in which the suspension functor is an equivalence of the homotopy category with itself. The prototypica

Subterminal object

In category theory, a branch of mathematics, a subterminal object is an object X of a category C with the property that every object of C has at most one morphism into X. If X is subterminal, then the

Grothendieck universe

In mathematics, a Grothendieck universe is a set U with the following properties: 1.
* If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.) 2.

Pulation square

In category theory, a branch of mathematics, a pulation square (also called a Doolittle diagram) is a diagram that is simultaneously a pullback square and a pushout square. It is a self-dual concept.

Nodal decomposition

In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism is a representation of as a product , where is a strong epimorphism, a bimorphism, and a strong monomorphis

Envelope (category theory)

In Category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech c

Globular set

In category theory, a branch of mathematics, a globular set is a higher-dimensional generalization of a directed graph. Precisely, it is a sequence of sets equipped with pairs of functions such that

F-algebra

In mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified ele

Eckmann–Hilton argument

In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two unital magma structures on a set where one is a homomorphism for the other.

Quotient of an abelian category

In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category by a Serre subcategory is the abelian category which, intuitively, is obtained from by ignoring (i.

Topological category

In category theory, a discipline in mathematics, the notion of topological category has a number of different, inequivalent definitions. In one approach, a topological category is a category that is e

Cartesian monoidal category

In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any ca

Compact object (mathematics)

In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition.

Distributive category

In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects , the canonical map is an isomorphism, and for all objects , the ca

Categorical trace

In category theory, a branch of mathematics, the categorical trace is a generalization of the trace of a matrix.

Double groupoid

In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.

Fiber functor

No description available.

Simplicially enriched category

In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial cate

Permutation category

In mathematics, the permutation category is a category where 1.
* an object is a natural number, 2.
* a morphism is an element of the symmetric group when and is none otherwise. It is equivalent as

Opetope

In category theory, a branch of mathematics, an opetope, a portmanteau of "operation" and "polytope", is a shape that captures higher-dimensional substitutions. It was introduced by John C. Baez and J

Adhesive category

In mathematics, an adhesive category is a category where pushouts of monomorphisms exist and work more or less as they do in the category of sets. An example of an adhesive category is the category of

Universal property

In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be use

Injective cogenerator

In category theory, a branch of mathematics, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approxima

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