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**.

The category **Set** is
complete and co-complete. The
product in this category is given by the
cartesian product of sets. The
coproduct is given by the
disjoint union: 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; 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).

**Set** is not
abelian,
additive nor
preadditive.

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*.

Another solution is to assume the existence of
Grothendieck 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 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. 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, 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.^{
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The same issues arise with other concrete categories, such as the category of groups or the category of topological spaces.

- Blass, A. (1984).
"The interaction between category theory and set theory" (PDF).
*Mathematical Applications of Category Theory*. Contemporary Mathematics. Vol. 30. American Mathematical Society. pp. 5–29. doi: 10.1090/conm/030/749767. ISBN 978-0-8218-5032-9. - Feferman, S. (1969).
"Set-theoretical foundations of category theory".
*Mac Lane 1969*. pp. 201–247. doi: 10.1007/BFb0059148. - Lawvere, F.W. An elementary theory of the category of sets (long version) with commentary
- Mac Lane, S. (2006) [1969]. "One universe as a foundation for category theory". In Mac Lane, S. (ed.).
*Reports of the Midwest Category Seminar III*. Lecture Notes in Mathematics. Vol. 106. Springer. pp. 192–200. doi: 10.1007/BFb0059147. ISBN 978-3-540-36150-3. -
Mac Lane, Saunders (September 1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics. Vol. 5. Springer. ISBN 0-387-98403-8. - Pareigis, Bodo (1970),
*Categories and functors*, Pure and applied mathematics, vol. 39, Academic Press, ISBN 978-0-12-545150-5