mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B...

category of sets, whose objects are sets and whose arrows are functions. Category theory is a branch of mathematics that seeks to generalize all of mathematics...

In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi...

required to be a set; a category where Hom(X, Y) is a set for all objects X and Y is called locally small. Because hom-sets may not be sets, some people prefer...

In category theory, a small set is one in a fixed universe of sets (as the word universe is used in mathematics in general). Thus, the category of small...

used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also...

mathematics, the category Ord has preordered sets as objects and order-preserving functions as morphisms. This is a category because the composition of two order-preserving...

ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial...

mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) R : A → B in this category is a relation...

Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set)....

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 Relative...

category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category...

pair of objects to a set of morphisms. So in the category of sets, this is an object of the category itself. In the same vein, in a closed category, the...

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas...

Nerve of a category Universal set, the notion of a 'set of all sets' Kashiwara, Masaki; Schapira, Pierre (2006). Categories and sheaves. empty category at...

Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the...

some fixed ring. In the category of sets, the pullback of functions f : X → Z and g : Y → Z always exists and is given by the set X × Z Y = { ( x , y )...

Look up set in Wiktionary, the free dictionary. Set, The Set, SET or SETS may refer to: Set (mathematics), a collection of elements Category of sets, the...

specializes Barr's definition to the case V = Set of ordinary categories, those whose homobjects form sets (of morphisms). Barr's monograph includes an appendix...

function. As a result, the empty set is the unique initial object of the category of sets and functions. The empty set can be turned into a topological...

(2012). "Types, Sets and Categories" (PDF). In Kanamory, Akihiro (ed.). Sets and Extensions in the Twentieth Century. Handbook of the History of Logic. Vol...

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any...

In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces...

thereby specified subset of Y {\displaystyle Y} . Example: In Set {\displaystyle \operatorname {Set} } , the category of sets and functions, the canonical...

Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition...

{\displaystyle D\colon C\to J} . In the special case when J is Set, the category of sets and functions, D is called a presheaf on C. Presheaves (over a...

In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in...

paradox concerns the impossibility of a set of sets, whose members are all sets that do not contain themselves. If such a set could exist, it could neither...

closed categories include: The category Set of all sets, with functions as morphisms, is Cartesian closed. The product X×Y is the Cartesian product of X and...

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...