# Category of sets

In the 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* are the total functions from *A* to *B*, and the composition of morphisms is the composition of functions.

Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind.

The axioms of a category are satisfied by **Set** because composition of functions is associative, and because every set *X* has an identity function id_{X} : *X* → *X* which serves as identity element for function composition.

The epimorphisms in **Set** are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.

The empty set serves as the initial object in **Set** with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in **Set**.

**Set** is the prototype of a concrete category; other categories are concrete if they are "built on" **Set** in some well-defined way.

Every two-element set serves as a subobject classifier in **Set**. 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*. **Set** is thus a topos (and in particular cartesian closed and exact in the sense of Barr).

Every non-empty set is an injective object in **Set**. Every set is a projective object in **Set** (assuming the axiom of choice).

The finitely presentable objects in **Set** are the finite sets. Since every set is a direct limit of its finite subsets, the category **Set** is a locally finitely presentable category.

If *C* is an arbitrary category, the contravariant functors from *C* to **Set** are often an important object of study. If *A* is an object of *C*, then the functor from *C* to **Set** that sends *X* to Hom_{C}(*X*,*A*) (the set of morphisms in *C* from *X* to *A*) is an example of such a functor. If *C* is a small category (i.e. the collection of its objects forms a set), then the contravariant functors from *C* to **Set**, together with natural transformations as morphisms, form a new category, a functor category known as the category of presheaves on *C*.

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. 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 like **Set** whose collection of objects forms a proper class are known as large 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 as NBG set theory. 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*.

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, 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, 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.^{[1]}^{[2]}^{[3]}

The same issues arise with other concrete categories, such as the category of groups or the category of topological spaces.