Dual (category theory)In , a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category Cop. Duality, as such, is the assertion that truth is invariant under this operation on statements.
Diagonal functorIn , a branch of mathematics, the diagonal functor is given by , which maps as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the : a product is a universal arrow from to . The arrow comprises the projection maps. More generally, given a , one may construct the , the objects of which are called . For each object in , there is a constant diagram that maps every object in to and every morphism in to .
Pushout (category theory)In , a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an P along with two morphisms X → P and Y → P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square.
Field of fractionsIn abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of an integral domain is sometimes denoted by or , and the construction is sometimes also called the fraction field, field of quotients, or quotient field of .
Free productIn mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from G ∗ H to K. Unless one of the groups G and H is trivial, the free product is always infinite.
Rng (algebra)In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng (IPA: rʊŋ) is meant to suggest that it is a ring without i, that is, without the requirement for an identity element. There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see ).
Sober spaceIn mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point. Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. All except the definition in terms of nets are described in. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the T0 axiom.
CurryingIn mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function that takes three arguments creates a nested unary function , so that the code gives the same value as the code or called in sequence, In a more mathematical language, a function that takes two arguments, one from and one from , and produces outputs in by currying is translated into a function that takes a single argument from and produces as outputs functions from to This is a natural one-to-one correspondence between these two types of functions, so that the sets together with functions between them form a .
Existential quantificationIn predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain.
Equaliser (mathematics)In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equaliser of f and g is the set of elements x of X such that f(x) equals g(x) in Y. Symbolically: The equaliser may be denoted Eq(f, g) or a variation on that theme (such as with lowercase letters "eq").
