Category of sets

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Definition

The category of sets, denoted , is defined as follows:

  • The objects of this category are sets.
  • For any two sets , is defined as the set of all functions from to .
  • 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.

Relation with other categories

Categories with functors to this 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.

Categories with functors from this category

  • 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.

Properties

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