Category of sets: Difference between revisions
(New page: ==Definition== The '''category of sets''', denoted <math>\operatorname{Set}</math>, is defined as follows: * The objects of this category are sets. * For any two sets <math>A,B</math...) |
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The category of sets is a [[locally small category]]. | 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== | |||
{{locally small}} | |||
Latest revision as of 00:11, 10 December 2008
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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.