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 In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...
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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
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
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
, 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 In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one ...
,''
:(ii)
''has
nodal decomposition In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism \varphi:X\to Y is a representation of \varphi as a product \varphi=\sigma\circ\beta\circ\pi, where \pi is a strong epimorphism, \beta a bimorphism, and \ ...
,''
:(iii)
''is co-well-powered in the class
,''
[A category is said to be ''co-well-powered in the class of epimorphisms'' , if for each object the category of all morphisms in going from is skeletally small.]
:(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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
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 In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...
of a Tikhonov
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is an envelope of
in the category
of all Tikhonov spaces in the class
of
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
s 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 In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is a ...
is an envelope of
in the category
of all stereotype algebras in the class
of all ''dense'' epimorphisms
[A morphism (i.e. a continuous unital homomorphism) of stereotype algebras is called dense if its set of values is dense in . ] in
with respect to the class
of all Banach algebras:
: 5. The ''smooth envelope''
of a
stereotype algebra In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is a ...
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 In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is a ...
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 In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is a ...
is a continuous envelope of
;
* for each
locally compact abelian group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the ...
the
Fourier transform is a continuous envelope of the
stereotype group algebra of measures with compact support on
.
In
abstract harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ...
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 algebra In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is a ...
s (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
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
,
differential geometry, and
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
– for certain classes of (not necessarily commutative) topological groups considered in these disciplines (
affine algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...
s, and some classes of
Lie groups and Moore groups).
See also
*
Refinement
Notes
References
*
*
*
*
*
*
*
*
*
{{Category theory
Category theory
Duality theories
Functional analysis