Enriched category

In category theory , the concept of the enriched category is a generalization of the concept of the locally small category .

In locally small categories one has a set of morphisms for every two objects , i.e. one object in . The basic idea of ​​enriched categories is that instead of other categories, it should be possible to use other categories for the morphism sets. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle X,Y\in {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}(X,Y)}$ ${\displaystyle \mathbf {Set} }$ ${\displaystyle \mathbf {Set} }$

For example, it is sometimes useful to view the morphism sets as topological spaces, i.e. as objects in TOP . In general, any monoidal categories can be used to define enriched categories.

definition

${\displaystyle {\mathcal {V}}}$is a monoidal category whose structure by monoidal and the arrow families , , is given. ${\displaystyle I\in {\mathcal {V}},\otimes \colon {\mathcal {V}}\times {\mathcal {V}}\to {\mathcal {V}}}$ ${\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C{\stackrel {\cong }{\longrightarrow }}A\otimes (B\otimes C)}$ ${\displaystyle \lambda _{A}\colon I\otimes A{\stackrel {\cong }{\longrightarrow }}A}$ ${\displaystyle \rho _{A}\colon A\otimes I{\stackrel {\cong }{\longrightarrow }}A}$

An over enriched category ${\displaystyle {\mathcal {V}}}$ or category has now ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {C}}}$

• Objects , ${\displaystyle X,Y,Z,\dotsc \in {\mathcal {C}}}$
• for every two objects one object that serves as a set of morphisms, ${\displaystyle X,Y\in {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}(X,Y)\in {\mathcal {V}}}$
• for each object an arrow in , which is intended to represent the identity arrow in , and ${\displaystyle X\in {\mathcal {C}}}$ ${\displaystyle i_{X}\colon I\to {\mathcal {C}}(X,X)}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {C}}}$
• for every three objects an arrow in , which is intended for the representation of the composition in . ${\displaystyle X,Y,Z\in {\mathcal {C}}}$ ${\displaystyle c_{X,Y,Z}\colon {\mathcal {C}}(Y,Z)\otimes {\mathcal {C}}(X,Y)\to {\mathcal {C}}(X,Z)}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {C}}}$

(Indices an are omitted below if it is for readability.) ${\displaystyle \alpha ,\lambda ,\rho ,i,c,\mathrm {id} }$

The following applies to all suitable indices:

• ${\displaystyle c\circ (i\otimes \mathrm {id} )=\lambda }$,
• ${\displaystyle c\circ (\mathrm {id} \otimes i)=\rho }$,
• ${\displaystyle c\circ (c\otimes \mathrm {id} )=c\circ (\mathrm {id} \otimes c)\circ \alpha }$.

Examples and special cases

• Ordinary locally small categories are -categories, where the monoidal structure is given by the Cartesian product . ${\displaystyle \mathbf {Set} }$ ${\displaystyle \mathbf {Set} }$
• Preadditive categories are -categories, where is the category of Abelian groups, with the tensor product of Abelian groups as a monoidal structure. ${\displaystyle \mathbf {Ab} }$ ${\displaystyle \mathbf {Ab} }$
• The category with two objects and exactly one arrow that is not an identity arrow has all finite products. -Categories are quasi-orders . ${\displaystyle \mathbf {2} :=\{0\to 1\}}$ ${\displaystyle \mathbf {2} }$
• The partial order of the nonnegative real numbers becomes a monoidal category or with the addition or the formation of the maximum . -Categories are then generalized metric spaces and -categories are generalized ultrametric spaces. The symmetry of the distance function, as well as the property that points must be identical with the distance , are not required. ${\displaystyle (\mathbb {R} ^{+},\geq )}$ ${\displaystyle \mathbb {R} _{+}^{+}}$ ${\displaystyle \mathbb {R} _{\max }^{+}}$ ${\displaystyle \mathbb {R} _{+}^{+}}$ ${\displaystyle \mathbb {R} _{\max }^{+}}$ ${\displaystyle 0}$
• For some , a category is itself , or can be understood as such. For example, this is the case for the category of Abelian groups , whose morphisms are Abelian groups with pointwise addition, or for the category of topological spaces , whose morphisms are topological spaces with the compact open topology . Such are called monoidally closed . If the monoidal structure is that of the Cartesian product , Cartesian is complete . ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$
• There is exactly one morphism object for a category with exactly one object . This is a monoid object in . ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle *}$ ${\displaystyle {\mathcal {C}}(*,*)}$ ${\displaystyle {\mathcal {V}}}$

Further definitions

V-functors

${\displaystyle {\mathcal {C,D}}}$are categories with or as identities and compositions. A function consists of ${\displaystyle {\mathcal {V}}}$ ${\displaystyle i,c}$ ${\displaystyle j,d}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$

• an object map that assigns an object of to each object of , and ${\displaystyle F\colon X\mapsto FX}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$
• a family of arrows in . ${\displaystyle F_{X,Y}\colon {\mathcal {C}}(X,Y)\to {\mathcal {D}}(FX,FY)}$ ${\displaystyle {\mathcal {V}}}$

If the indices are omitted, the following applies: ${\displaystyle F}$

• ${\displaystyle F\circ i=j}$,
• ${\displaystyle d\circ (F\otimes F)=F\circ c}$.

Natural transformations

${\displaystyle {\mathcal {C,D}}}$are categories with or as identities and compositions. be -functions. The usual definition of natural transformations can be adapted to categories. A natural transformation must specify an arrow for each object that is the component of . It must then for everyone ${\displaystyle {\mathcal {V}}}$ ${\displaystyle i,c}$ ${\displaystyle j,d}$ ${\displaystyle F,G\colon {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle \psi \colon F\to G}$ ${\displaystyle X\in {\mathcal {C}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle \psi _{X}\colon I\to {\mathcal {D}}(FX,GX)}$ ${\displaystyle X}$ ${\displaystyle \psi }$ ${\displaystyle X,Y\in {\mathcal {C}}}$

${\displaystyle d\circ \psi _{X}\circ \rho ^{-1}\circ G=d\circ \psi _{Y}\circ \lambda ^{-1}\circ F\colon {\mathcal {C}}(X,Y)\to {\mathcal {D}}(FX,GY)}$

be valid.

It is also possible to define an object of natural transformations . This is an object in , namely the end ${\displaystyle F\to G}$ ${\displaystyle {\mathcal {V}}}$

${\displaystyle E:=\int _{X\in {\mathcal {C}}}{\mathcal {D}}(FX,GX)}$.

"Elements" of , ie arrows , then represent natural transformations and result from composition with the projections of their components. ${\displaystyle E}$ ${\displaystyle I\to E}$ ${\displaystyle E}$

literature

• GM Kelly: Basic Concepts of Enriched Category Theory . In: Lecture Notes in Mathematics 64 . Cambridge University Press, 1982 ( mta.ca [accessed May 30, 2014]).