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):