In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function It is the most basic and the most commonly used category in mathematics.
The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property
The empty set serves as the initial object in Set with empty functions as morphisms. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Mathematics, an empty function is a function whose domain is the Empty set. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. In Mathematics, a singleton is a set with exactly one element There are thus no zero objects in Set.
The category Set is complete and co-complete. In Mathematics, a complete category is a category in which all small limits exist The product in this category is given by the cartesian product of sets. 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 of mathematics such as Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. The coproduct is given by the disjoint union: given sets Ai where i ranges over some index set I, we construct the coproduct as the union of Ai×{i} (the cartesian product with i serves to insure that all the components stay disjoint). In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated
Set is the prototype of a concrete category; other categories are concrete if they "resemble" Set in some well-defined way. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving
Every two-element set serves as a subobject classifier in Set. In Category theory, a subobject classifier is a special object &Omega of a category intuitively the Subobjects of an object X correspond to the morphisms The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) Set is thus a topos (and in particular cartesian closed). In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets In Category theory, a category is cartesian closed if roughly speaking any Morphism defined on a product of two objects can be naturally identified with a morphism
Set is not abelian, additive or preadditive; it does not even have zero morphisms. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Category theory, a zero morphism is a special kind of "trivial" Morphism.
Every not initial object in Set is injective and (assuming the axiom of choice) also projective. In Mathematics, especially in the field of Category theory, the concept of injective object is a generalization of the concept of Injective module. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation
The size of the category of sets
It is often assumed that the collection of all sets is not a set. For instance, this follows from the axiom of foundation. In mathematics the axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory and was introduced by. This leads to problems when formalizing what the category Set really is.
One way to resolve this difficulty is to say that the collection of all sets is a proper class. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously One can then work in a framework, such as NBG set theory, that distinguishes between sets and classes. In the Foundations of mathematics, Von Neumann–Bernays–Gödel set theory ( NBG) is an Axiomatic set theory that is a Conservative extension The category Set is then said to be large.
Another solution is to assume the existence of Grothendieck universes. In Mathematics, a Grothendieck universe is a set U with the following properties If x is an element of U and if y The objects of Set are then sets relative to some universe, U. The collection of all these sets is again a set, but not a set in U.
Various other solutions, and variations on the above, have been proposed[1][2][3].
References
- Mac Lane, Saunders (September 1998). Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Categories for the Working Mathematician. Springer. ISBN 0-387-98403-8. (Volume 5 in the series Graduate Texts in Mathematics)
- ^ Mac Lane, S. Graduate Texts in Mathematics (GTM is a series of graduate-level Textbooks in Mathematics published by Springer-Verlag. One universe as a foundation for category theory. Springer Lect. Notes Math. 106 (1969): 192–200.
- ^ Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math. 106 (1969): 201–247.
- ^ Blass, A. The interaction between category theory and set theory. Contemporary Mathematics 30 (1984).
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