Envelope (category theory)

In :Category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called refinement.

Definition 

Suppose  is a category,  an object in , and  and  two classes of morphisms in . The definition of an envelope of  in the class  with respect to the class  consists of two steps.

 A morphism  in  is called an extension of the object  in the class of morphisms  with respect to the class of morphisms , if , and for any morphism  from the class  there exists a unique morphism  in  such that .

 
 An extension  of the object  in the class of morphisms  with respect to the class of morphisms  is called an envelope of  in  with respect to , if for any other extension  (of  in  with respect to ) there is a unique morphism  in  such that . The object  is also called an envelope of  in  with respect to .
 
Notations:

In a special case when  is a class of all morphisms whose ranges belong to a given class of objects  in  it is convenient to replace  with  in the notations (and in the terms):
 

Similarly, if  is a class of all morphisms whose ranges belong to a given class of objects  in  it is convenient to replace  with  in the notations (and in the terms):
 

For example, one can speak about an envelope of  in the class of objects  with respect to the class of objects :

Nets of epimorphisms and functoriality 

Suppose that to each object  in a category  it is assigned a subset  in the class
 of all epimorphisms of the category , going from , and the following three requirements are fulfilled:
 for each object  the set  is non-empty and is directed to the left with respect to the pre-order inherited from 
 
 for each object  the covariant system of morphisms generated by   
 
has a colimit  in , called the local limit in ;
 for each morphism  and for each element  there are an element  and a morphism  such that 
 
Then the family of sets  is called a net of epimorphisms in the category .

Examples.
 For each locally convex topological vector space  and for each closed convex balanced neighbourhood of zero  let us consider its kernel  and the quotient space  endowed with the normed topology with the unit ball , and let  be the completion of  (obviously,  is a Banach space, and it is called the quotient Banach space of  by ). The system of natural mappings  is a net of epimorphisms in the category  of locally convex topological vector spaces.  
 For each locally convex topological algebra  and for each submultiplicative closed convex balanced neighbourhood of zero , 
 ,
 let us again consider its kernel  and the quotient algebra  endowed with the normed topology with the unit ball , and let  be the completion of  (obviously,  is a Banach algebra, and it is called the quotient Banach algebra of  by ). The system of natural mappings  is a net of epimorphisms in the category  of locally convex topological algebras.  

Theorem. Let  be a net of epimorphisms in a category  that generates a class of morphisms  on the inside:
 
Then for any class of epimorphisms  in , which contains all local limits ,
 
the following holds:
(i) for each object  in  the local limit  is an envelope  in  with respect to : 
 
(ii) the envelope  can be defined as a functor.

Theorem. Let  be a net of epimorphisms in a  category  that generates a class of morphisms  on the inside:
 
Then for any monomorphically complementable class of epimorphisms  in  such that  is co-well-powered in  the envelope  can be defined as a functor.

Theorem.
Suppose a category  and a class of objects  have the following properties:
(i)  is cocomplete,
(ii)  has nodal decomposition,
(iii)  is co-well-powered in the class ,
(iv)  goes from :
 ,
(v)  differs morphisms on the outside: for any two different parallel morphisms  there is a morphism  such that ,
(vi)  is closed with respect to passage to colimits,
(vii)  is closed with respect to passage from the codomain of a morphism to its nodal image: if , then .
Then the envelope  can be defined as a functor.

Examples 
In the following list all envelopes can be defined as functors.
 1. The completion  of a locally convex topological vector space  is an envelope of  in the category  of all locally convex spaces with respect to the class  of Banach spaces: . Obviously,  is the inverse limit of the quotient Banach spaces  (defined above):

 2. The Stone–Čech compactification  of a Tikhonov topological space  is an envelope of  in the category  of all Tikhonov spaces in the class  of compact spaces with respect to the same class : 
 3. The Arens-Michael envelope  of a locally convex topological  algebra  with a separately continuous multiplication is an envelope of  in the category  of all (locally convex) topological algebras (with separately continuous multiplications) in the class  with respect to the class  of Banach algebras: . The algebra  is the inverse limit of the quotient Banach algebras  (defined above):

 4. The holomorphic envelope  of a stereotype algebra  is an envelope of  in the category  of all stereotype algebras in the class  of all dense epimorphisms in  with respect to the class  of all Banach algebras: 
 5. The smooth envelope  of a stereotype algebra  is an envelope of  in the category  of all involutive stereotype algebras in the class  of all dense epimorphisms in  with respect to the class  of all differential homomorphisms into various C*-algebras with joined self-adjoined nilpotent elements: 
 6. The continuous envelope  of a stereotype algebra  is an envelope of  in the category  of all involutive stereotype algebras in the class  of all dense epimorphisms in  with respect to the class  of all C*-algebras:

Applications 

Envelopes appear as standard functors in various fields of mathematics. Apart from the examples given above, 

 the Gelfand transform  of a commutative involutive stereotype algebra  is a continuous envelope of ; 

 for each locally compact abelian group  the Fourier transform   is a continuous envelope of the stereotype group algebra  of measures with compact support on . 

In abstract harmonic analysis the notion of envelope plays a key role in the generalizations of the Pontryagin  duality theory to the classes of non-commutative groups: the holomorphic, the smooth and the continuous envelopes of stereotype algebras (in the  examples given above) lead respectively to the constructions of the holomorphic, the smooth and the continuous dualities in big geometric disciplines – complex geometry, differential geometry, and topology – for certain classes of (not necessarily commutative) topological groups considered in these disciplines (affine algebraic groups, and some classes of Lie groups and Moore groups).

See also
Refinement

Notes

References 

 

 

Category theory
Duality theories
Functional analysis