In the mathematical Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions field of category theory In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows, the category of sets, denoted as Set, is the category In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may be abstract entities of any kind whose objects In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows are sets A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In. The arrows or morphisms In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures between sets A and B are all functions The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain from A to B. Care must be taken in the definition of Set to avoid set theoretic Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics paradoxes.

The category of sets is the most basic and the most commonly used category in mathematics. Many other categories (such as the category of groups In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory, with group homomorphisms From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that as arrows) add additional structure to the objects of this category and/or restrict the arrows to functions of a particular kind.

Properties of the category of sets

The epimorphisms Epimorphisms are analogues of surjective functions, but they are not exactly the same. The dual of an epimorphism is a monomorphism in Set are the surjective In mathematics, a function is said to be surjective or onto if its image is equal to its codomain. A function f: X → Y is surjective if and only if for every y in the codomain Y there is at least one x in the domain X such that f = y. A surjective function is called a surjection. Surjections are sometimes denoted by a two-headed rightwards arrow, maps, the monomorphisms In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation are the injective In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain. If in addition all of the elements in the codomain are in fact mapped to by maps, and the isomorphisms In abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f ∠are the bijective In mathematics, a bijection, or a bijective function, is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y and no unmapped element exists in either X or Y maps.

The empty set In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set serves as the initial object In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object : T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called in Set with empty functions as morphisms. Every singleton In mathematics, a singleton is a set with exactly one element. For example, the set {0} is a singleton is a terminal object In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object : T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object : T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called in Set.

The category Set is complete and co-complete. The product In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the & in this category is given by the cartesian product In mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept of sets. The coproduct In category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each is given by the disjoint union The elements of the disjoint union are ordered pairs . Here i serves as an auxiliary index that indicates which Ai the element x came from. Each of the sets Ai is canonically embedded in the disjoint union as the set: given sets A_i where i ranges over some index set I, we construct the coproduct as the union of A_i×{i} (the cartesian product with i serves to ensure that all the components stay disjoint).

Set is the prototype of a concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete; other categories are concrete if they "resemble" Set in some well-defined way.

Every two-element set serves as a subobject classifier In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements belong to the subobject in question. Because of in Set. The power object of a set A is given by its power set In mathematics, given a set S, the power set of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set, and the exponential object In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. An exponential object may also be called a power object or map object of the sets A and B is given by the set of all functions from A to B. Set is thus a topos In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory (and in particular cartesian closed 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 defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that they provide a natural setting for lambda).

Set is not abelian, additive In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C or preadditive In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom in C has the structure of an abelian group, and composition of morphisms is bilinear over the integers. Its zero morphisms In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object are the empty functions ∅ → X.^[1]

Every not initial object in Set is injective and (assuming the axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins and there is no "rule" for which object to) also projective In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module . Various equivalent characterizations of these modules are available.

Foundations for the category of sets

In Zermelo–Fraenkel set theory the collection of all sets is not a set; this follows from the axiom of foundation. One refers to collections that are not sets as proper classes In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF set theory, the notion of class is informal, whereas other set theories, such as NBG set. One can't handle proper classes as one handles sets; in particular, one can't write that those proper classes belong to a collection (either a set or a proper class). This is a problem: it means that the category of sets cannot be formalized straightforwardly in this setting.

One way to resolve the problem is to work in a system that gives formal status to proper classes, such as NBG set theory In the foundations of mathematics, Von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes proper classes, objects having members but that. In this setting, categories formed from sets are said to be small and those (like Set) that are formed from proper classes are said to be large.

Another solution is to assume the existence of Grothendieck universes A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. As an example, we will prove an easy proposition. Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for instance if a set belongs to a universe, its elements and its powerset will belong to the universe). The existence of Grothendieck universes (other than the empty set and the set V_ω of all hereditarily finite sets) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence of strongly inaccessible cardinals In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable. Assuming this extra axiom, one can limit the objects of Set to the elements of a particular universe. (There is no "set of all sets" within the model, but one can still reason about the class U of all inner sets, i. e., elements of U.)

In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes. (This is necessarily a proper class In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF set theory, the notion of class is informal, whereas other set theories, such as NBG set, but each Grothendieck universe is a set because it is an element of some larger Grothendieck universe.) However, one does not work directly with the "category of all sets". Instead, theorems are expressed in terms of the category Set_U whose objects are the elements of a sufficiently large Grothendieck universe U, and are then shown not to depend on the particular choice of U. As a foundation for category theory In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows, this approach is well matched to a system like Tarski-Grothendieck set theory in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all Set_U but not of Set.

Various other solutions, and variations on the above, have been proposed.^[2][3][4]

The same issues arise with other concrete categories, such as the category of groups In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory or the category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory.

See also

• Set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics
• Small set (category theory)

Notes

1. ^ Section I.7 of Pareigis 1970
2. ^ Mac Lane 1969
3. ^ Feferman 1969
4. ^ Blass 1984

References

• Blass, A. The interaction between category theory and set theory. Contemporary Mathematics 30 (1984).
• Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math. 106 (1969): 201–247.
• Lawvere, F.W. An elementary theory of the category of sets (long version) with commentary
• Mac Lane, S. One universe as a foundation for category theory. Springer Lect. Notes Math. 106 (1969): 192–200.
• Mac Lane, Saunders Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg (September 1998). Categories for the Working Mathematician. Springer. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0-387-98403-8. (Volume 5 in the series Graduate Texts in Mathematics Graduate Texts in Mathematics is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a standard size. This particular series is easily identified by a white band at the top of the book)
• Pareigis, Bodo (1970), Categories and functors, Pure and applied mathematics, 39, Academic Press Academic Press is an academic book publisher that is now part of the Elsevier Publishing Company, ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 978-0-125-45150-5

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