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 formula_1 is a category, formula_2 an object in formula_1, and formula_4 and formula_5 two classes of morphisms in formula_1. The definition of an envelope of formula_2 in the class formula_4 with respect to the class formula_5 consists of two steps.
Notations:
formula_39
In a special case when formula_4 is a class of all morphisms whose ranges belong to a given class of objects formula_41 in formula_1 it is convenient to replace formula_4 with formula_41 in the notations (and in the terms):
formula_45
Similarly, if formula_5 is a class of all morphisms whose ranges belong to a given class of objects formula_47 in formula_1 it is convenient to replace formula_5 with formula_47 in the notations (and in the terms):
formula_51
For example, one can speak about an "envelope of formula_2 in the class of objects formula_41 with respect to the class of objects formula_47":
formula_55
Nets of epimorphisms and functoriality.
Suppose that to each object formula_56 in a category formula_57 it is assigned a subset formula_58 in the class
formula_59 of all epimorphisms of the category formula_57, going from formula_2, and the following three requirements are fulfilled:
Then the family of sets formula_77. Obviously, formula_78 is the inverse limit of the quotient Banach spaces formula_79 (defined above):
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 (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).