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.
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 :
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 :
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:
Then the family of sets is called a net of epimorphisms in the category.
Examples.
  1. 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 . The system of natural mappings is a net of epimorphisms in the category of locally convex topological vector spaces.
  2. For each locally convex topological algebra and for each submultiplicative closed convex balanced neighbourhood of zero,
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:
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:
Then the envelope can be defined as a functor.

Examples

In the following list all envelopes can be defined as functors.

Applications

Envelopes appear as standard functors in various fields of mathematics. Apart from the examples given above,
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 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 topological groups considered in these disciplines.