In themathematical field ofcategory theory, thecategory of sets, denoted bySet, is thecategory whoseobjects aresets. The arrows ormorphisms between setsA andB are thefunctions fromA toB, and the composition of morphisms is thecomposition of functions.
Many other categories (such as thecategory of groups, withgroup homomorphisms as arrows) add structure to the objects of the category of sets or restrict the arrows to functions of a particular kind (or both).
The axioms of a category are satisfied bySet because composition of functions isassociative, and because every setX has anidentity function idX :X →X which serves as identity element for function composition.
Theepimorphisms inSet are thesurjective maps, themonomorphisms are theinjective maps, and theisomorphisms are thebijective maps.
Theempty set serves as theinitial object inSet withempty functions as morphisms. Everysingleton is aterminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus nozero objects inSet.
The categorySet iscomplete and co-complete. Theproduct in this category is given by thecartesian product of sets. Thecoproduct is given by thedisjoint union: given setsAi wherei ranges over some index setI, we construct the coproduct as the union ofAi×{i} (the cartesian product withi serves to ensure that all the components stay disjoint).
Set is the prototype of aconcrete category; other categories are concrete if they are "built on"Set in some well-defined way.
Every two-element set serves as asubobject classifier inSet. The power object of a setA is given by itspower set, and theexponential object of the setsA andB is given by the set of all functions fromA toB.Set is thus atopos (and in particularcartesian closed andexact in the sense of Barr).
Set is notabelian,additive norpreadditive.
Every non-empty set is aninjective object inSet. Every set is aprojective object inSet (assuming theaxiom of choice).
Thefinitely presentable objects inSet are the finite sets. Since every set is adirect limit of its finite subsets, the categorySet is alocally finitely presentable category.
IfC is an arbitrary category, thecontravariant functors fromC toSet are often an important object of study. IfA is an object ofC, then the functor fromC toSet that sendsX to HomC(X,A) (the set of morphisms inC fromX toA) is an example of such a functor. IfC is asmall category (i.e. the collection of its objects forms a set), then the contravariant functors fromC toSet, together with natural transformations as morphisms, form a new category, afunctor category known as the category ofpresheaves onC.
InZermelo–Fraenkel set theory the collection of all sets is not a set; this follows from theaxiom of foundation. One refers to collections that are not sets asproper classes. One cannot handle proper classes as one handles sets; in particular, one cannot write that those proper classes belong to a collection (either a set or a proper class). This is a problem because it means that the category of sets cannot be formalized straightforwardly in this setting. Categories likeSet whose collection of objects forms a proper class are known aslarge categories, to distinguish them from the small categories whose objects form a set.
One way to resolve the problem is to work in a system that gives formal status to proper classes, such asNBG set theory. In this setting, categories formed from sets are said to besmall and those (likeSet) that are formed from proper classes are said to belarge.
Another solution is to assume the existence ofGrothendieck universes. 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 of allhereditarily finite sets) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence ofstrongly inaccessible cardinals. Assuming this extra axiom, one can limit the objects ofSet to the elements of a particular universe. (There is no "set of all sets" within the model, but one can still reason about the classU of all inner sets, i.e., elements ofU.)
In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes. (This is necessarily aproper class, 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 categorySetU whose objects are the elements of a sufficiently large Grothendieck universeU, and are then shown not to depend on the particular choice ofU. As a foundation forcategory theory, this approach is well matched to a system likeTarski–Grothendieck set theory in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of allSetU but not ofSet.
Various other solutions, and variations on the above, have been proposed.[1][2][3]
The same issues arise with other concrete categories, such as thecategory of groups or thecategory of topological spaces.