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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 K is a category, X an object in K, and \Omega and \Phi two classes of morphisms in K. The definition of an envelope of X in the class \Omega with respect to the class \Phi consists of two steps. * A morphism \sigma:X\to X' in K is called an ''extension of the object X in the class of morphisms \Omega with respect to the class of morphisms \Phi'', if \sigma\in\Omega, and for any morphism \varphi:X\to B from the class \Phi there exists a unique morphism \varphi':X'\to B in K such that \varphi=\varphi'\circ\sigma. * An extension \rho:X\to E of the object X in the class of morphisms \Omega with respect to the class of morphisms \Phi is called an ''envelope of X in \Omega with respect to \Phi'', if for any other extension \sigma:X\to X' (of X in \Omega with respect to \Phi) there is a unique morphism \upsilon:X'\to E in K such that \rho=\upsilon\circ\sigma. The object E is also called an ''envelope of X in \Omega with respect to \Phi''. Notations: \rho=\text_^X, \qquad E=\text_^X. In a special case when \Omega is a class of all morphisms whose ranges belong to a given class of objects L in K it is convenient to replace \Omega with L in the notations (and in the terms): \rho=\text_^L X, \qquad E=\text_^L X. Similarly, if \Phi is a class of all morphisms whose ranges belong to a given class of objects M in K it is convenient to replace \Phi with M in the notations (and in the terms): \rho=\text_M^ X, \qquad E=\text_M^ X. For example, one can speak about an ''envelope of X in the class of objects L with respect to the class of objects M'': \rho=\text_M^L X, \qquad E=\text_M^L X.


Nets of epimorphisms and functoriality

Suppose that to each object X\in\operatorname() in a category it is assigned a subset ^X in the class \operatorname^X of all epimorphisms of the category , going from X, and the following three requirements are fulfilled: * for each object X the set ^X is non-empty and is directed to the left with respect to the pre-order inherited from \operatorname^X :: \forall \sigma,\sigma'\in ^X\quad \exists\rho\in^X\quad \rho\to\sigma\ \& \ \rho\to\sigma', * for each object X the covariant system of morphisms generated by ^X :: \ :has a colimit \varprojlim ^X in K, called the ''local limit'' in X; * for each morphism \alpha:X\to Y and for each element \tau\in^Y there are an element \sigma\in^X and a morphism \alpha_\sigma^\tau:\operatorname\sigma\to\operatorname\tau such that :: \tau\circ\alpha=\alpha_\sigma^\tau\circ\sigma. 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 ...
X and for each closed convex balanced neighbourhood of zero U\subseteq X let us consider its kernel \operatornameU=\bigcap_\varepsilon\cdot U and the quotient space X/\operatornameU endowed with the normed topology with the unit ball U+\operatornameU, and let X/U=(X/\operatornameU)^\blacktriangledown be the completion of X/\operatornameU (obviously, X/U is a Banach space, and it is called the ''quotient Banach space'' of X by U). The system of natural mappings X\to X/U is a net of epimorphisms in the category \text of locally convex topological vector spaces. # For each locally convex topological algebra A and for each ''submultiplicative'' closed convex balanced neighbourhood of zero U\subseteq X, :::: U\cdot U\subseteq U, :: let us again consider its kernel \operatornameU=\bigcap_\varepsilon\cdot U and the quotient algebra A/\operatornameU endowed with the normed topology with the unit ball U+\operatornameU, and let A/U=(A/\operatornameU)^\blacktriangledown be the completion of A/\operatornameU (obviously, A/U 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 X by U). The system of natural mappings A\to A/U is a net of epimorphisms in the category \text of locally convex topological algebras. Theorem. ''Let be a net of epimorphisms in a category that generates a class of morphisms \varPhi on the inside:'' :: \subseteq \varPhi \subseteq \operatorname() \circ . ''Then for any class of epimorphisms \varOmega in K, which contains all local limits \varprojlim ^X,'' :: \\subseteq\varOmega\subseteq\operatorname(K), ''the following holds:'' :(i) ''for each object X in the local limit \varprojlim ^X is an envelope'' \operatorname_\varPhi^\varOmega X in \varOmega with respect to \varPhi: :: \varprojlim ^X=\operatorname_\varPhi^\varOmega X, :(ii) ''the envelope \operatorname_\varPhi^\varOmega can be defined as a functor.'' Theorem. ''Let be a net of epimorphisms in a category that generates a class of morphisms \varPhi on the inside:'' :: \subseteq \varPhi \subseteq \operatorname() \circ . ''Then for any monomorphically complementable class of epimorphisms \varOmega in K such that K is co-well-powered in \varOmega the envelope \operatorname_\varPhi^\varOmega can be defined as a functor.'' Theorem. ''Suppose a category K and a class of objects L have the following properties:'' :(i) K ''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) K ''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) K ''is co-well-powered in the class \operatorname,''A category K is said to be ''co-well-powered in the class of epimorphisms'' \operatorname, if for each object X the category \operatorname^X of all morphisms in \operatorname going from X is skeletally small. :(iv) \operatorname(K,L) ''goes from K:'' :: \forall X\in\operatorname(K)\quad \exists \varphi\in\operatorname(K)\quad \operatorname\varphi=X\quad\&\quad\operatorname\varphi\in L, :(v) L ''differs morphisms on the outside: for any two different parallel morphisms \alpha\ne\beta:X\to Y there is a morphism \varphi:Y\to Z\in L such that \varphi\circ\alpha\ne\varphi\circ\beta,'' :(vi) L ''is closed with respect to passage to colimits,'' :(vii) L ''is closed with respect to passage from the codomain of a morphism to its nodal image: if \operatorname\alpha\in L, then \operatorname_\infty\alpha\in L.'' ''Then the envelope \operatorname_L^L can be defined as a functor.''


Examples

In the following list all envelopes can be defined as functors. : 1. The ''completion'' X^\blacktriangledown 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 ...
X is an envelope of X in the category \text of all locally convex spaces with respect to the class \text of Banach spaces: X^\blacktriangledown=\text_^X. Obviously, X^\blacktriangledown is the inverse limit of the quotient Banach spaces X/U (defined above): ::::X^\blacktriangledown=\lim_X/U. : 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 ...
\beta:X\to\beta X 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 ...
X is an envelope of X in the category \text of all Tikhonov spaces in the class \text 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 \text: \beta X=\text_^X. : 3. The ''Arens-Michael envelope'' A^ of a locally convex topological algebra A with a separately continuous multiplication is an envelope of A in the category \text of all (locally convex) topological algebras (with separately continuous multiplications) in the class \text with respect to the class \text of Banach algebras: A^= \text_^A . The algebra A^ is the inverse limit of the quotient Banach algebras A/U (defined above): ::::A^=\lim_A/U. : 4. The ''holomorphic envelope'' \text_ A 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 ...
A is an envelope of A in the category \text of all stereotype algebras in the class \text of all ''dense'' epimorphismsA morphism (i.e. a continuous unital homomorphism) of stereotype algebras \varphi:A\to B is called dense if its set of values \varphi(A) is dense in B. in \text with respect to the class \text of all Banach algebras: \text_ A= \text_^A. : 5. The ''smooth envelope'' \text_ A 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 ...
A is an envelope of A in the category \text of all involutive stereotype algebras in the class \text of all ''dense'' epimorphisms in \text with respect to the class \text of all differential homomorphisms into various C*-algebras with joined self-adjoined nilpotent elements: \text_ A= \text_^A. : 6. The ''continuous envelope'' \text_ A 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 ...
A is an envelope of A in the category \text of all involutive stereotype algebras in the class \text of all ''dense'' epimorphisms in \text with respect to the class \text^* of all C*-algebras: \text_ A= \text_^A.


Applications

Envelopes appear as standard functors in various fields of mathematics. Apart from the examples given above, * the Gelfand transform G_A:A\to C(\text A) 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 ...
A is a continuous envelope of A; * 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 ...
G the Fourier transform F_A:C^\star(G)\to C(\widehat) is a continuous envelope of the stereotype group algebra C^\star(G) of measures with compact support on G. 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