(mathematics) transformed into an abelian group

(mathematics, category theory) A functor related to another functor by an adjunction.

(category theory) One of a pair of functors such that the domain and codomain of one of them are identical to the codomain and domain of the other one, respectively, and such that there is a pair of natural transformations which turns the pair of functors into an adjunction.

(category theory) One of a pair of morphisms which relate to each other through a pair of adjoint functors.

(category theory) Given a pair of categories 𝒞 and 𝒟: an anti-parallel pair of functors F:𝒞→𝒟 and G:𝒟→𝒞 and a natural transformation 𝜂: mbox id_C→GF called “unit” such that for any object A∈𝒞, for any object B∈𝒟, and for any morphism f:A→GB, there is a unique morphism g:FA→B such that Gg∘𝜂_A=f. (Note: there is another natural transformation called “counit” as well but its existence may be derived by theorem.) The pair of functors express a similarity between the pair of categories which is weaker than that of an equivalence of categories.

(algebraic geometry) A group that is an algebraic variety.

(mathematics, category theory) A category that retains some of the structure of the category of binary relations between sets, representing a high-level generalisation of that category.

(computer science) a generalization of the list-producing unfolds known from functional programming to arbitrary abstract data types that can be described as final coalgebras

(category theory) A category in which every bimorphism is an isomorphism.

(category theory) A cartesian closed category which also has an initial object and such that for any pair of objects, A and B, in the category, the category has another object which is their coproduct, A∐B.

(mathematics) A particular construct in category theory

(mathematics) A functor between bicategories

(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.

(category theory) A morphism which is both a monomorphism and an epimorphism.

(category theory) A biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct.

(category theory) A category which has a terminal object and which for every two objects A and B has a product A × B and an exponential object Bᴬ.

(category theory) A natural transformation whose naturality squares are pullbacks.

(category theory) A pullback.

(category theory) For a given category C, its dual category Cᵒᵖ is obtained by reversing the direction of each one of the arrows of C.

(category theory) A procedure that defines theorems in terms of category theory by mapping concepts from set theory to category theory.

(mathematics) A collection of objects, together with a transitively closed collection of composable arrows between them, such that every object has an identity arrow, and such that arrow composition is associative.

(group theory) The theorem stating that every group G is isomorphic to a subgroup of the symmetric group acting on G.

(linear algebra) an eigenvector

(category theory) A morphism from an object to the subobject classifier which corresponds to a unique subobject of the said object, which subobject is the pullback, along this morphism, of the "true" global element of the subobject classifier.

(mathematics) A natural transformation from the identity functor to a certain endofunctor.

(mathematics) A full embedding whose image is a cocomplete category and for which every functor with a cocomplete image has an extension to a cocontinuous functor that is unique up to natural isomorphism.

(category theory) An object V together with an arrow going from each object of a diagram to V such that for any arrow A in the diagram, the pair of arrows to V which form a triangle with A also commute with it.

(category theory) A morphism which yields the same morphism when pre-composed with either one of a parallel pair of morphisms.

(mathematics, of a functor) Commutative with small colimits; right-exact and transforming small direct sums into small direct sums.

(category theory) An arrow whose domain is the codomain of a parallel pair of arrows and which forms part of the colimit of that parallel pair.

A contravariant functor.

(mathematics) A particular linear subspace in a coalgebra

(category theory) The identity element of a cogroup.

(category theory) The inversion operation of a cogroup object. A coinversion must satisfy the dual versions of the axioms for group objects.

(category theory, informal) cokernel

(category theory) For a category with zero morphisms: the coequalizer between a given morphism and the zero morphism which is parallel to that given morphism.

(category theory) The cocone of a diagram through which any other cocone of that same diagram can factor uniquely.

(mathematics) The conjecture that the process of taking a number n and, if n is even, dividing it by 2, or if n is odd, multiplying it by 3 and adding 1, then repeating this process, will eventually yield 1.

(linear algebra) The transpose of the row echelon form of the transpose of a given matrix.

(category theory) A category built out of a pair of functors that have the same codomain.

(mathematics, of a diagram of morphisms) Such that any two sequences of morphisms with the same initial and final positions compose to the same morphism.

(mathematics, category theory) A monad of the opposite category.

(category theory) An object V together with an arrow going from V to each object of a diagram such that for any arrow A in the diagram, the pair of arrows from V which subtend A also commute with it. (Then V can be said to be the cone’s vertex and the diagram which the cone subtends can be said to be its base.)

(category theory) A morphism which yields the same morphism when post-composed with either one of a parallel pair of morphisms.

(category theory) A functor which reverses composition.

(category theory, of a functor) which reverses composition

(category theory) A functor which maps a morphism f:X → Y to a morphism F(f):F(Y) → F(X), such that if h=g∘f, then F(h)=F(f)∘F(g).

(category theory) The dual of a presheaf.

(mathematics) A certain mathematical object in connection with Leibniz algebras.

(mathematics) The left inverse of a morphism.

(category theory) The category opposite to that of a slice category.

(category theory) Describing an object in a category, such that there is precisely one morphism that maps that object to every object in the category.

(mathematics) Of an abelian group: whose every extension by a torsion-free group splits.

(mathematics) In an adjunction, a natural transformation from the composition of the left adjoint functor with the right adjoint functor to the identity functor of the domain of the right adjoint functor.

(mathematics) A principle that gives a condition for finding the eigenvalues for a real symmetric matrix.

(category theory) (Of a functor) which preserves composition.

(category theory) A morphism from an object to the product of that object with itself, which morphism is induced by a pair of identity morphisms of the said object.

(category theory) A functor from an index category to another category. The objects and morphisms of the index category need not have any internal substance, but rather merely outline the connective structure of at least some part of the diagram's codomain. If the index category is J and the codomain is C, then the diagram is said to be "of type J in C".

(category theory) a colimit

(category theory) A high-level generalization of the preceding that applies to objects in an arbitrary category and produces a new object constructable by morphisms from each of the original objects.

(algebra) A set of algebraic structures (which are part of a concrete category; e.g., modules) and a set of morphisms between them (e.g., module homomorphisms) which all together form a small category which is the image of a covariant functor whose domain is a directed poset.

(category theory) A category whose morphisms are all identity morphisms.

(topology) Synonym of cover

(category theory) Being the dual of some other category; containing the same objects but with source and target reversed for all morphisms.

(linear algebra) row echelon form

(linear algebra) A square matrix which is obtained from the identity matrix by a single elementary row (or column) operation.

(category theory) A Cartesian closed category which has a subobject classifier.

(mathematics) A bifunctor between a bicategory and itself.

(category theory) A functor that maps a category to itself.

(category theory, of a morphism) That is an epimorphism.

(category theory) A morphism p such that for any other pair of morphisms f and g, if f∘p=g∘p, then f = g.

(category theory) Said of a morphism: to pre-compose with each of a parallel pair of morphisms so as to yield the same composite morphism.

(category theory) A morphism whose codomain is the domain of a parallel pair of morphisms and which forms part of the limit of that parallel pair. Equivalently, a morphism which equalizes a parallel pair of morphisms in a limiting way, which is to say that any other morphism which equalizes that parallel pair factors through this limiting morphism; and moreover such factorization is unique.

(category theory) An adjunction whose unit and counit are both natural isomorphisms.

(geometry) A differential operator that allows a contravariant to be constructed from an invariant.

(category theory) An object which indexes a family of arrows between two given objects in a universal way, meaning that any other indexed family of arrows between the same given pair of objects must factor uniquely through this universally-indexed family of arrows.

(category theory) Said of a morphism: to be equal to a composition of two morphisms, one of which is the given morphism (that is being said to be "factored through").

(category theory) A functor which maps morphisms from its source to its target category in such a way that the restriction of that mapping to any source hom-set is injective into the corresponding target hom-set.

(category theory) The pullback of a morphism along a global element (called the fiber of the morphism over the global element).

(category theory) Said to be of a morphism over a global element: The pullback of the said morphism along the said global element.

(category theory) A structure made of two categories, two functors from the first to the second category, and a transformation from one of the functors to the other.

(category theory) A category that is induced by a multidigraph thus: it has as its objects the vertices of the multidigraph and its morphisms are paths in the multidigraph; composition of morphisms is concatenation of paths, as long as the end of one path coincides with the beginning of the other path; an identity morphism of an object is an “empty path” at that vertex.

(category theory) A functor which maps morphisms from its source to its target category in such a way that the restriction of that mapping to any source hom-set is surjective into the corresponding target hom-set.

(category theory) A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows, in such a way as to preserve domains and codomains (of the arrows) as well as composition and identities.

(category theory) A category whose objects are functors (from some fixed given domain category to some fixed given codomain category) and whose morphisms are natural transformations.

(category theory) Of or pertaining to a functor

(topology) For a specified topological space, the group whose elements are homotopy classes of loops (images of some arbitrary closed interval whose endpoints are both mapped to a designated point) and whose group operation is concatenation.

(category theory, order theory) A type of correspondence between partially ordered sets (posets), also applicable to preordered sets.

(algebra) A finite field.

(mathematics) An iterative method used to solve a linear system of equations.

(group theory) For given field F and order n, the group of invertible n×n matrices, with the group operation of matrix multiplication.

(category theory) A morphism whose codomain is some specified object.

(category theory) The identity element of an object when thought of as a generalized element of that object.

(category theory) A morphism from the terminal object to a given object (to which it is said to belong).

(set theory) A fuzzy kind of category in which membership is determined not in binary (true-or-false) fashion but by a more general interval.

(category theory, of a morphism f) A morphism 𝛤_f from the domain of f to the product of the domain and codomain of f, such that the first projection applied to 𝛤_f equals the identity of the domain, and the second projection applied to 𝛤_f is equal to f.

(mathematics) An abelian group constructed from a commutative monoid in a universal way.

(category theory, algebraic geometry) A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.

(category theory) Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element.

(category theory, scheme theory) A group object that is an object in a category of schemes; a scheme that has certain properties that generalise the concept of group.

(algebra and category theory) A set with a partial binary operation that is associative and has identities and inverses.

(category theory) A functor from a category to itself which maps each object of that category to itself and each morphism of that category to itself.

(linear algebra) A diagonal matrix all of the diagonal elements of which are equal to 1.

(category theory) A unique morphism corresponding to each object of a category, which has its domain equal to its codomain, and which composed with any morphism (with which it is composable) gives that same morphism.

(category theory) An object within a category which sends out arrows to all other objects in that category, and such that each of these arrows is unique.

(category theory) A morphism from either one of the two components of a coproduct to that coproduct.

(category theory) The pullback of a corner of monics.

(category theory) A morphism which is both a left inverse and a right inverse.

(category theory) a limit

(algebra) A set of algebraic structures (which are part of a concrete category; e.g., groups) and a set of morphisms between them (e.g., group homomorphisms) which all together form a small category which is the image of a contravariant functor whose domain is a directed poset.

(algebraic geometry, category theory) An epimorphism of group schemes that is surjective and has a finite kernel.

(category theory) A morphism which has a two-sided inverse; the composition of the morphism and such an inverse yields either one of two identity morphisms (depending on the order of composition).

(category theory) A construct that generalizes the notion of extending a function's domain of definition.

(algebra) A method of constructing a Lie algebra from a Jordan algebra.

(mathematics, category theory) For a category with zero morphisms: the equalizer of a given morphism and the zero morphism which is parallel to that given morphism.

(category theory) A couple of morphisms which constitute the pullback of a given morphism along itself.

(category theory) A category naturally associated to any monad T, and equivalent to the category of free T-algebras.

(category theory) For a given morphism f : X → Y, its right inverse (if it has one) is a morphism s : Y → X such that s∘f= mbox id_X.

(category theory) The cone of a diagram through which any other cone of that same diagram can factor uniquely.

(group theory) Any group of invertible matrices over a specified field, with the group operation of matrix multiplication.

(algebra, ring theory) An ideal which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure).

(mathematics) A nonabelian norm

(category theory) A monoid object in the category of endofunctors of a fixed category.

(category theory) Of a morphism: that it is a monomorphism.

(category theory) A category 𝒞 with a bifunctor ⊗:𝒞⨯𝒞→𝒞 which may be called tensor product, an associativity isomorphism 𝛼_(A,B,C):(A⊗B)⊗C≃A⊗(B⊗C), an object I which may be called tensor unit, a left unit natural isomorphism 𝜆_A:I⊗A≃A, a right unit natural isomorphism 𝜌_A:A⊗I≃A, and some "coherence conditions" (pentagon and triangle commutative diagrams for those isomorphisms).

(category theory) A morphism n such that for any other morphisms f and g, if n∘f=n∘g then f = g.

(mathematics, category theory) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.

(category theory) hom-set

(mathematics) A generalization of the concept of category that allows morphisms of multiple arity.

(algebra) Closed under submodules, direct sums, and injective hulls.

(category theory) A natural transformation whose every component is an isomorphism.

(category theory) An object which has a distinguished global element (which may be called z, for “zero”) and a distinguished endomorphism (which may be called s, for “successor”) such that iterated compositions of s upon z (i.e., sⁿ∘z) yields other global elements of the same object which correspond to the natural numbers (sⁿ∘z↔n). Such object has the universal property that for any other object with a distinguished global element (call it z’) and a distinguished endomorphism (call it s’), there is a unique morphism (call it φ) from the given object to the other object which maps z to z’ (𝜙∘z=z') and which commutes with s; i.e., 𝜙∘s=s'∘𝜙.

(category theory) A morphism between a pair of parallel functors such that if each object of the shared domain category subtends a correlated arrow — called a component — in the shared codomain (which arrow represents the difference between applying the second functor and the first functor to the correlated object) then each arrow of the shared domain subtends a commuting square — called a naturality square — between two components (correlated to the domain and codomain of the arrow).

(linear algebra) any other answers apart from ⃑x=⃑0 to the linear system A⃑x=⃑0, signifying linear dependence of the linear system

(category theory) An object that is both an initial and terminal object.

(category theory) An instance of one of the two kinds of entities that form a category, the other kind being the arrows (also called morphisms).

(category theory) Any functor into a specified base category.

(mathematics) A morphism from a diagonal overfunctor to the base category of the overcategory.

(category theory) A morphism of an overcategory.

(mathematics) A ring for which a specified subring R contains no nonempty socles and for which any R-module is of finite length.

(mathematics) An object whose dual space is some other specified object.

(category theory) A category together with its inversion.

(category theory, algebraic geometry) A contravariant functor whose domain is a category whose objects are open sets of a topological space and whose morphisms are inclusion mappings. The functorial images of the open sets are sets of things called sections which are said to be "over" those open sets. The (contravariant) functorial images of those inclusion mappings are functions which are called restrictions.

(category theory) The category c is called a prestack over a category C with a Grothendieck topology if it is fibered over C and for any object U of C and objects x, y of c with image U, the functor from objects over U to sets taking F:V→U to Hom(F*x,F*y) is a sheaf.

(algebra, field theory) An element that generates a simple extension.

(topology, category theory) A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space.

(mathematics) A generalization of a relation in category theory

(category theory) A morphism from a categorical product to one of its (two) components.

(mathematics) A mapping between categories that is just like a functor except that f(x∘y)=f(x)∘f(y) and f(1)=1 do not hold as exact equalities but only up to coherent isomorphisms.

(category theory) Within a Cartesian square (which has a pair of divergent morphisms and a pair of convergent morphisms) the divergent morphism which is directly opposite to a given one of the convergent morphisms, said to be “along” the convergent morphism which is between that pair of opposite morphisms. (The pullback is said to be “of” the given morphism.)

(category theory) The colimit of a pair of morphisms which share the same domain.

(category theory) A subobject of a product of objects.

(category theory) A functor from some category to the category of sets (Set) which is naturally isomorphic to a hom functor.

(category theory) For a given morphism f : X → Y, its right inverse (if it has one) is a morphism s : Y → X such that f∘s= mbox id_Y.

(algebra) A function from one ring to another one which preserves the structure of the additive operation, the structure of the multiplicative operation, the zero element (additive identity), and, if there is one, the unity element (multiplicative identity).

(category theory) A generalization of an association scheme from the point of view of small categories.

(category theory) A right inverse.

(functional analysis) Being a Hilbert space operator which is Hermitian and also whose eigenvectors span the entire Hilbert space.

(mathematics) injective with itself (in a higher dimension)

(mathematics) an eigenvalue problem that would be a positone eigenvalue problem except that the nonlinear function is not positive when its argument is zero.

(algebra) A finite (and thereby algebraic) extension of a base field such that every element of the extension is the root of a separable polynomial over the base field.

(category theory) A collection of morphisms in a category whose codomain is a certain fixed object of that category, which collection is closed under precomposition by any morphism in the category.

(category theory) A formal specification of a mathematical structure or a data type described in terms of a graph and diagrams (and cones (and cocones)) on it. It can be implemented by means of “models”, which are functors which are graph homomorphisms from the formal specification to categories such that the diagrams become commutative, the cones become limiting (i.e., products), the cocones become colimiting (i.e., sums).

(category theory) A category whose objects are morphisms (of some given category) with a common codomain, and whose morphisms are commuting triangles where two morphisms of each of such triangles share the said common codomain.

(category theory) A category such that all of its objects form a set and all of its morphisms form a set.

(category theory) A morphism which has a right inverse.

(category theory) A morphism which has a left inverse.

(category theory) A subclass of a category which is itself a category, whose arrows are a restriction of the arrows of the parent category, and whose composition rule is a restriction of the parent category's

(category theory) A functor such that all of the objects it maps are mapped by the parent functor, and for any arrow it maps the parent functor includes the same mapping (although it may also map arrows from the same domain to additional images outside the image of the subfunctor).

(category theory) An object and a monomorphism from it to another object, which monomorphism is interpreted as an inclusion. Actually it is an equivalence class of monomorphisms to the same object, where the equivalence relation is the ability of a pair of monomorphisms to factor through each other.

(category theory) An object which serves as the codomain of a classifying morphism, together with a "true" global element of the said object.

(set theory) A function for which every element of the codomain is mapped to by some element of the domain; (formally) Any function f:X→Y for which for every y∈Y, there is at least one x∈X such that f(x)=y.

(category theory) An object within a category which receives arrows from all other objects in that category, and such that each of these arrows is unique.

(mathematics) A natural isomorphism of a functor (F) such that F(x⊗y)→F(y⊗x).

(linear algebra) The process of rearranging elements in a matrix, by interchanging their respective row and column positional indicators.

(mathematics) A higher-dimensional form of category

(category theory) An isomorphism between a pair of products which have the same components but in swapped order, which isomorphism commutes with the two associated product diagrams.

(category theory) a forgetful functor

(algebra) A set upon which some operations are defined, thereby yielding an algebraic structure.

(category theory) In an adjunction, a natural transformation from the identity functor of the domain of the left adjoint functor to the composition of the right adjoint functor with the left adjoint functor.

(category theory) The terminal object of a comma category from a functor to a fixed object; or, dually, the initial object of a comma category from a fixed object to a functor.

(mathematics) The property of there being a unique morphism from an arbitrary object to the universal morphism of a comma category which is from a functor to a fixed object; or, dually, the property of there being a unique morphism from the universal morphism to an arbitrary object of a comma category which is from a fixed object to a functor. (Caveat: the uniqueness is up to isomorphism.)

(category theory) Said of a category or topos: that it has a terminal object (or that it has no zero object, if it is a topos) and in which any distinct pair of parallel morphisms can be distinguished by their distinct compositions with a global element of their domain; i.e. any pair of parallel morphisms is distinct if and only if there is a global element in their domain that does not equalize them.

(category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.

(category theory) An embedding of a category 𝒞 within the category of functors from 𝒞 to the category of sets. Such an embedding is effected by a Yoneda functor.

(category theory) A functor from a given category to the category of functors from that given category to Set (the category of sets) which maps any object of the given category to a hom functor represented by that object and any morphism to a natural isomorphism induced uniquely by that morphism according to the Yoneda lemma.

(category theory) Given a category 𝒞 with an object A, let H be a hom functor represented by A, and let F be any functor (not necessarily representable) from 𝒞 to Sets, then there is a natural isomorphism between Nat(H,F), the set of natural transformations from H to F, and the set F(A). (Any natural transformation 𝛼 from H to F is determined by what 𝛼_A( mbox id_A) is.)

(category theory) A morphism which is both a constant morphism and a coconstant morphism.

(category theory) An object which is both an initial object and a terminal object.