Envelope (category theory) explained

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

\Omega

and

\Phi

two classes of morphisms in

. The definition of an envelope of

in the class

\Omega

with respect to the class

\Phi

consists of two steps.

A morphism

\sigma:X\toX'

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\toB

from the class

\Phi

there exists a unique morphism

\varphi':X'\toB

such that

\varphi=\varphi'\circ\sigma

An extension

\rho:X\toE

of the object

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\toX'

(of

\Omega

with respect to

\Phi

) there is a unique morphism

\upsilon:X'\toE

such that

\rho=\upsilon\circ\sigma

. The object

is also called an envelope of
X

in
\Omega

with respect to
\Phi

. Notations:

	\Omega
\rho=env
	\Phi

	\Omega
E=Env
	\Phi

In a special case when

\Omega

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

it is convenient to replace

\Omega

with

in the notations (and in the terms):

	L
\rho=env
	\Phi

	L
E=Env
	\Phi

Similarly, if

\Phi

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

it is convenient to replace

\Phi

with

in the notations (and in the terms):

	\Omega
\rho=env
	M

	\Omega
E=Env
	M

For example, one can speak about an envelope of

in the class of objects

with respect to the class of objects

	L
\rho=env
	M

	L
E=Env
	M

Nets of epimorphisms and functoriality

Suppose that to each object

X\in\operatorname{Ob}({K})

in a category

{K}

it is assigned a subset

{lN}^X

in the class

\operatorname{Epi}^X

of all epimorphisms of the category

{K}

, going from

, and the following three requirements are fulfilled:

for each object

the set

{lN}^X

is non-empty and is directed to the left with respect to the pre-order inherited from

\operatorname{Epi}^X

\forall\sigma,\sigma'\in{lN}^X\exists\rho\in{lN}^{X
\rho\to\sigma \&} \rho\to\sigma',

for each object

the covariant system of morphisms generated by

{lN}^X

	\sigma; \rho,\sigma\in{lN}
\{\iota
	\rho

^{X, \rho\to\sigma\}}

has a colimit

\varprojlim{lN}^X

, called the local limit in

;

for each morphism

\alpha:X\toY

and for each element

\tau\in{lN}^Y

there are an element

\sigma\in{lN}^X

and a morphism

	\tau:\operatorname{Cod}\sigma\to\operatorname{Cod}\tau
\alpha
	\sigma

^[1] such that

	\tau\circ\sigma.
\tau\circ\alpha=\alpha
	\sigma

Then the family of sets

{lN}=\{{lN}^{X; X\in\operatorname{Ob}({K})\}}

is called a net of epimorphisms in the category

{K}

Examples.

and for each closed convex balanced neighbourhood of zero

U\subseteqX

let us consider its kernel

\operatorname{Ker}U=cap_{\varepsilon>0}\varepsilon ⋅ U

and the quotient space

X/\operatorname{Ker}U

endowed with the normed topology with the unit ball

U+\operatorname{Ker}U

, and let

X/U=(X/\operatorname{Ker}U)^{\blacktriangledown}

be the completion of

X/\operatorname{Ker}U

(obviously,

X/U

is a Banach space, and it is called the quotient Banach space of

). The system of natural mappings

X\toX/U

is a net of epimorphisms in the category

LCS

of locally convex topological vector spaces.

For each locally convex topological algebra

and for each submultiplicative closed convex balanced neighbourhood of zero

U\subseteqX

U ⋅ U\subseteqU

let us again consider its kernel

\operatorname{Ker}U=cap_{\varepsilon>0}\varepsilon ⋅ U

and the quotient algebra

A/\operatorname{Ker}U

endowed with the normed topology with the unit ball

U+\operatorname{Ker}U

, and let

A/U=(A/\operatorname{Ker}U)^{\blacktriangledown}

be the completion of

A/\operatorname{Ker}U

(obviously,

A/U

is a Banach algebra, and it is called the quotient Banach algebra of

). The system of natural mappings

A\toA/U

is a net of epimorphisms in the category

LCS

of locally convex topological algebras.

Theorem. Let

{lN}

be a net of epimorphisms in a category

{K}

that generates a class of morphisms

\varPhi

on the inside:

{lN}\subseteq\varPhi\subseteq\operatorname{Mor}({K})\circ{lN}.

Then for any class of epimorphisms
\varOmega

in
K

, which contains all local limits
\varprojlim{lN}^X

,

\{\varprojlim{lN}^X; X\in\operatorname{Ob}(K)\}\subseteq\varOmega\subseteq\operatorname{Epi}(K),

the following holds:

(i) for each object

{K}

the local limit

\varprojlim{lN}^X

is an envelope
\varOmega
\operatorname{env}
\varPhi

X

in
\varOmega

with respect to
\varPhi

:

\varprojlim

	\varOmega
{lN}
	\varPhi

(ii) the envelope

	\varOmega
\operatorname{Env}
	\varPhi

can be defined as a functor.

Theorem. Let

{lN}

be a net of epimorphisms in a category

{K}

that generates a class of morphisms

\varPhi

on the inside:

{lN}\subseteq\varPhi\subseteq\operatorname{Mor}({K})\circ{lN}.

Then for any monomorphically complementable class of epimorphisms
\varOmega

in
K

such that
K

is co-well-powered^[2] in
\varOmega

the envelope
\varOmega
\operatorname{Env}
\varPhi

can be defined as a functor.

Theorem.Suppose a category

and a class of objects

have the following properties:

(i)

is cocomplete,

(ii)

has nodal decomposition,

(iii)

is co-well-powered in the class
\operatorname{Epi}

,^[3]

(iv)

\operatorname{Mor}(K,L)

goes from
K

:

\forallX\in\operatorname{Ob}(K) \exists\varphi\in\operatorname{Mor}(K) \operatorname{Dom}\varphi=X \& \operatorname{Cod}\varphi\inL

(v)

differs morphisms on the outside: for any two different parallel morphisms
\alpha\ne\beta:X\toY

there is a morphism
\varphi:Y\toZ\inL

such that
\varphi\circ\alpha\ne\varphi\circ\beta

,

(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
\operatorname{Cod}\alpha\inL

, then
\operatorname{Im}_{infty\alpha\in}L

.Then the envelope
L
\operatorname{Env}
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

is an envelope of

in the category

LCS

of all locally convex spaces with respect to the class

Ban

of Banach spaces:

	LCS
X
	Ban

. Obviously,

X^{\blacktriangledown}

is the inverse limit of the quotient Banach spaces

X/U

(defined above):

	\blacktriangledown=\lim
X
	0\getsU

X/U.

\beta:X\to\betaX

of a Tikhonov topological space

is an envelope of

in the category

Tikh

of all Tikhonov spaces in the class

Com

of compact spaces with respect to the same class

Com

\beta

	Com
X=Env
	Com

3. The Arens-Michael envelope

A^AM

of a locally convex topological algebra

with a separately continuous multiplication is an envelope of

in the category

TopAlg

of all (locally convex) topological algebras (with separately continuous multiplications) in the class

TopAlg

with respect to the class

Ban

of Banach algebras:

A^AM=

	TopAlg
Env
	Ban

. The algebra

A^AM

is the inverse limit of the quotient Banach algebras

A/U

(defined above):

A^AM=\lim_0\getsA/U.

4. The holomorphic envelope

Env_\calA

of a stereotype algebra

is an envelope of

in the category

SteAlg

of all stereotype algebras in the class

DEpi

of all dense epimorphisms^[4] in

SteAlg

with respect to the class

Ban

of all Banach algebras:

Env_\calA=

	DEpi
Env
	Ban

5. The smooth envelope

Env_\calA

of a stereotype algebra

is an envelope of

in the category

InvSteAlg

of all involutive stereotype algebras in the class

DEpi

of all dense epimorphisms in

InvSteAlg

with respect to the class

DiffMor

of all differential homomorphisms into various C*-algebras with joined self-adjoined nilpotent elements:

Env_\calA=

	DEpi
Env
	DiffMor

6. The continuous envelope

Env_\calA

of a stereotype algebra

is an envelope of

in the category

InvSteAlg

of all involutive stereotype algebras in the class

DEpi

of all dense epimorphisms in

InvSteAlg

with respect to the class

C^*

of all C*-algebras:

Env_\calA=

	DEpi
Env
	C^*

Applications

Envelopes appear as standard functors in various fields of mathematics. Apart from the examples given above,

G_A:A\toC(SpecA)

of a commutative involutive stereotype algebra

is a continuous envelope of

;

the Fourier transform

	\star(G)\to
F
	A:C

C(\widehat{G})

is a continuous envelope of the stereotype group algebra

C^\star(G)

of measures with compact support on

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).

References

Book: Helemskii, A.Ya. . Banach and locally convex algebras . Oxford Science Publications. 1993 . .
Pirkovskii. A.Yu.. Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras. Trans. Moscow Math. Soc.. 2008. 69. 27–104. 10.1090/S0077-1554-08-00169-6. free.
Akbarov. S.S.. Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity. Journal of Mathematical Sciences. 2009. 162. 4. 459–586. 0806.3205. 10.1007/s10958-009-9646-1. 115153766.
Akbarov . S.S. . 2010 . Stereotype algebras and duality for Stein groups . Moscow State University .
Akbarov. S.S.. Envelopes and refinements in categories, with applications to functional analysis. Dissertationes Mathematicae. 2016. 513. 1–188. 1110.2013. 10.4064/dm702-12-2015. 118895911.
Akbarov. S.S.. Continuous and smooth envelopes of topological algebras. Part 1. Journal of Mathematical Sciences. 2017a . 227. 5. 531–668. 1303.2424. 10.1007/s10958-017-3599-6. 126018582.
Akbarov. S.S.. Continuous and smooth envelopes of topological algebras. Part 2. Journal of Mathematical Sciences. 2017b . 227. 6. 669–789. 1303.2424. 10.1007/s10958-017-3600-4. 128246373.
Akbarov. S.S.. The Gelfand transform as a C*-envelope . Mathematical Notes. 2013. 94. 5–6. 814–815. 10.1134/S000143461311014X. 121354607.
Kuznetsova. Y.. A duality for Moore groups. Journal of Operator Theory. 2013. 69. 2. 101–130. 0907.1409. 2009arXiv0907.1409K. 10.7900/jot.2011mar17.1920. 115177410.

Notes and References

\operatorname{Cod}\varphi

means the codomain of the morphism
\varphi

.
A category
K

is said to be co-well-powered in a class of morphisms
\varOmega

, if for each object
X

the category
\varOmega^X

of all morphisms in
\varOmega

going from
X

is skeletally small.
A category
K

is said to be co-well-powered in the class of epimorphisms
\operatorname{Epi}

, if for each object
X

the category
\operatorname{Epi}^X

of all morphisms in
\operatorname{Epi}

going from
X

is skeletally small.
A morphism (i.e. a continuous unital homomorphism) of stereotype algebras
\varphi:A\toB

is called dense if its set of values
\varphi(A)

is dense in
B

.

Envelope (category theory) explained

Definition

Nets of epimorphisms and functoriality

Examples

Applications

See also

References

Notes and References