Category of sets

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Definition

The category of sets, denoted $\operatorname {Set}$ , is defined as follows:

The objects of this category are sets.
For any two sets $A,B$ , $\operatorname {Set} (A,B)$ is defined as the set of all functions from $A$ to $B$ .
The identity morphism from any object to itself is defined as the identity map on that object.
The composition of morphisms is defined by function composition.

The category of sets is a locally small category.

A category along with a faithful functor to the category of sets is termed a concrete category. Many basic categories studied in algebra and topology are concrete: they come equipped with obvious choices of faithful functors to the category of sets. The faithful functors in these cases are usually termed forgetful functors.
For any category, we can construct functors from that category to the category of sets. In fact, every object in the category gives rise to such a functor: this is called the covariant Hom functor. We can also construct contravariant functors to the category of sets: the contravariant Hom functor.

A typical kind of functor from the category of sets to other category is a free functor; these are usually right-adjoint functors to the forgetful functors.