Comma categoryIn mathematics, a comma category (a special case being a slice category) is a construction in . It provides another way of looking at morphisms: instead of simply relating objects of a to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some s and colimits.
CoequalizerIn , a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary . It is the categorical construction to the equalizer. A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y. More explicitly, a coequalizer of the parallel morphisms f and g can be defined as an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g.
Free algebraIn mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X1,...,Xn} is the free R-module with a basis consisting of all words over the alphabet {X1,...
Disjoint unionIn mathematics, a disjoint union (or discriminated union) of a family of sets is a set often denoted by with an injection of each into such that the of these injections form a partition of (that is, each element of belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union. In , the disjoint union is the coproduct of the , and thus defined up to a bijection. In this context, the notation is often used. The disjoint union of two sets and is written with infix notation as .
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").
Free categoryIn mathematics, the free category or path category generated by a directed graph or quiver is the that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next. More precisely, the objects of the category are the vertices of the quiver, and the morphisms are paths between objects. Here, a path is defined as a finite sequence where is a vertex of the quiver, is an edge of the quiver, and n ranges over the non-negative integers.
Product categoryIn the mathematical field of , the product of two C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors. The product category C × D has: as : pairs of objects (A, B), where A is an object of C and B of D; as arrows from (A1, B1) to (A2, B2): pairs of arrows (f, g), where f : A1 → A2 is an arrow of C and g : B1 → B2 is an arrow of D; as composition, component-wise composition from the contributing categories: (f2, g2) o (f1, g1) = (f2 o f1, g2 o g1); as identities, pairs of identities from the contributing categories: 1(A, B) = (1A, 1B).
Abuse of notationIn mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time). However, since the concept of formal/syntactical correctness depends on both time and context, certain notations in mathematics that are flagged as abuse in one context could be formally correct in one or more other contexts.
Essentially uniqueIn mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which is often formalized using an equivalence relation. A related notion is a universal property, where an object is not only essentially unique, but unique up to a unique isomorphism (meaning that it has trivial automorphism group).
