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.

The definition[1] of an envelope of

consists of two steps.

In a special case when

is a class of all morphisms whose ranges belong to a given class of objects

it is convenient to replace

in the notations (and in the terms):

is a class of all morphisms whose ranges belong to a given class of objects

it is convenient to replace

in the notations (and in the terms):

For example, one can speak about an envelope of

with respect to the class of objects

it is assigned a subset

, and the following three requirements are fulfilled: Then the family of sets

is called a net of epimorphisms in the category

be a net of epimorphisms in a category

that generates a class of morphisms

on the inside: Then for any class of epimorphisms

, which contains all local limits

, the following holds: Theorem.

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

can be defined as a functor.

have the following properties: Then the envelope

can be defined as a functor.

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

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[20] 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).