Enriched category

From Academic Kids

In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner.


We will define what it means for C to be an enriched category over a monoidal category M.

We require the following structures:

  • Let Ob(C) be a set (or proper class, if you prefer). Then an element of Ob(C) is an object of C.
  • For each pair (A,B) of objects of C, let Hom(A,B) be an object of M. Then Hom(A,B) is the hom-object of A and B.
  • For each object A of C, let idA be a morphism in M from I to Hom(A,A), where I is a fixed identity object of the monoidal operation of M. Then idA is the identity morphism of A.
  • For each triple (A,B,C) of objects of C, let ° be a morphism in M from Hom(B,C) ⊗ Hom(A,B) to Hom(A,C), where ⊗ is the monoidal operation in M. Then ° is the composition morphism of A, B, and C.

We require the following axioms:

  • Associativity: Given objects A, B, C, and D of C, we can go from Hom(C,D) ⊗ Hom(B,C) ⊗ Hom(A,B) to Hom(A,D) in two ways, depending on which composition we do first. These must give the same result.
  • Left identity: Given objects A and B of C, we can go from I ⊗ Hom(A,B) to just Hom(A,B) in two ways, either by using idB on I and then using composition, or by simply using the fact that I is an identity for ⊗ in M. These must give the same result.
  • Right identity: Given objects A and B of C, we can go from Hom(A,B) ⊗ I to just Hom(A,B) in two ways, either by using idA on I and then using composition, or by simply using the fact that I is an identity for ⊗ in M. These must give the same result.

We should include some commutative diagrams illustrating these axioms.

Then C (consisting of all the structures listed above) is a category enriched over M.


The most straightforward example is to take M to be a category of sets, with the Cartesian product for the monoidal operation. Then C is nothing but an ordinary category. If M is the category of small sets, then C is a locally small category, because the hom-sets will all be small. Similarly, if M is the category of finite sets, then C is a locally finite category.

If M is the category 2 with Ob(2) = {0,1}, a single nonidentity morphism (from 0 to 1), and ordinary multiplication of numbers as the monoidal operation, then C can be interpreted as a preordered set. Specifically, AB iff Hom(A,B) = 1.

If M is a category of pointed sets with Cartesian product for the monoidal operation, then C is a category with zero morphisms. Specifically, the zero morphism from A to B is the special point in the pointed set Hom(A,B).

If M is a category of abelian groups with tensor product as the monoidal operation, then C is a preadditive category.

A property

If there is a monoidal functor from a monoidal category M to a monoidal category N, then any category enriched over M can be reinterpreted as a category enriched over N. Every monoidal category M has a monoidal functor M(I, –) to the category of sets, so any enriched category has an underlying ordinary category. In many examples (such as those above) this functor is faithful, so a category enriched over M can be described as an ordinary category with certain additional structure or properties.


Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)


  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Personal tools