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.

## Contents

* 1 Definition
* 2 Nets of epimorphisms and functoriality
* 3 Examples
* 4 Applications
* 5 See also
* 6 Notes
* 7 References

## Definition

Suppose K is a category, X an object in K, and Ω and Φ two classes of morphisms in K. The definition[1] of an envelope of X in the class Ω with respect to the class Φ consists of two steps.

Extension.

* A morphism σ:X->X′ in K is called an extension of the object X in the class of morphisms Ω with respect to the class of morphisms Φ, if σ∈Ω, and for any morphism φ:X->B from the class Φ there exists a unique morphism φ′:X′->B in K such that φ=φ′∘σ.

Envelope.

* An extension ρ:X->E of the object X in the class of morphisms Ω with respect to the class of morphisms Φ is called an envelope of X in Ω with respect to Φ, if for any other extension σ:X->X′ (of X in Ω with respect to Φ) there is a unique morphism υ:X′->E in K such that ρ=υ∘σ. The object E is also called an envelope of X in Ω with respect to Φ.

Notations:

ρ=envΦΩX,E=EnvΦΩX.

In a special case when Ω is a class of all morphisms whose ranges belong to a given class of objects L in K it is convenient to replace Ω with L in the notations (and in the terms):

ρ=envΦLX,E=EnvΦLX.

Similarly, if Φ is a class of all morphisms whose ranges belong to a given class of objects M in K it is convenient to replace Φ with M in the notations (and in the terms):

ρ=envMΩX,E=EnvMΩX.

For example, one can speak about an envelope of X in the class of objects L with respect to the class of objects M:

ρ=envMLX,E=EnvMLX.

## Nets of epimorphisms and functoriality

Suppose that to each object X∈Ob⁡(K) in a category K it is assigned a subset 𝒩X in the class EpiX of all epimorphisms of the category K, going from X, and the following three requirements are fulfilled:

* for each object X the set 𝒩X is non-empty and is directed to the left with respect to the pre-order inherited from EpiX

∀σ,σ′∈𝒩X∃ρ∈𝒩Xρ->σ & ρ->σ′,

* for each object X the covariant system of morphisms generated by 𝒩X

{ιρσ; ρ,σ∈𝒩X, ρ->σ}
has a colimit lim<-𝒩X in K, called the local limit in X;

* for each morphism α:X->Y and for each element τ∈𝒩Y there are an element σ∈𝒩X and a morphism αστ:Cod⁡σ->Cod⁡τ[2] such that

τ∘α=αστ∘σ.

Then the family of sets 𝒩={𝒩X; X∈Ob⁡(K)} is called a net of epimorphisms in the category K.

Examples.

1. For each locally convex topological vector space X and for each closed convex balanced neighbourhood of zero U⊆X let us consider its kernel Ker⁡U=⋂ε>0ε⋅U and the quotient space X/Ker⁡U endowed with the normed topology with the unit ball U+Ker⁡U, and let X/U=(X/Ker⁡U)▾ be the completion of X/Ker⁡U (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->X/U is a net of epimorphisms in the category LCS of locally convex topological vector spaces.
2. For each locally convex topological algebra A and for each submultiplicative closed convex balanced neighbourhood of zero U⊆X,

U⋅U⊆U,
let us again consider its kernel Ker⁡U=⋂ε>0ε⋅U and the quotient algebra A/Ker⁡U endowed with the normed topology with the unit ball U+Ker⁡U, and let A/U=(A/Ker⁡U)▾ be the completion of A/Ker⁡U (obviously, A/U is a Banach algebra, and it is called the quotient Banach algebra of X by U). The system of natural mappings A->A/U is a net of epimorphisms in the category LCS of locally convex topological algebras.

Theorem.[3] Let 𝒩 be a net of epimorphisms in a category K that generates a class of morphisms Φ on the inside:

𝒩⊆Φ⊆Mor⁡(K)∘𝒩.

Then for any class of epimorphisms Ω in K, which contains all local limits lim<-𝒩X,

{lim<-𝒩X; X∈Ob⁡(K)}⊆Ω⊆Epi⁡(K),

the following holds:

(i) for each object X in K the local limit lim<-𝒩X is an envelope envΦΩX in Ω with respect to Φ:

lim<-𝒩X=envΦΩX,
(ii) the envelope EnvΦΩ can be defined as a functor.

Theorem.[4] Let 𝒩 be a net of epimorphisms in a category K that generates a class of morphisms Φ on the inside:

𝒩⊆Φ⊆Mor⁡(K)∘𝒩.

Then for any monomorphically complementable class of epimorphisms Ω in K such that K is co-well-powered[5] in Ω the envelope EnvΦΩ can be defined as a functor.

Theorem.[6] Suppose a category K and a class of objects L have the following properties:

(i) K is cocomplete,
(ii) K has nodal decomposition,
(iii) K is co-well-powered in the class Epi,[7]
(iv) Mor⁡(K,L) goes from K:

∀X∈Ob⁡(K)∃φ∈Mor⁡(K)Dom⁡φ=X&Cod⁡φ∈L,
(v) L differs morphisms on the outside: for any two different parallel morphisms α≠β:X->Y there is a morphism φ:Y->Z∈L such that φ∘α≠φ∘β,
(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 Cod⁡α∈L, then Im∞α∈L.

Then the envelope EnvLL can be defined as a functor.

## Examples

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

1\. The completion X▾ of a locally convex topological vector space X is an envelope of X in the category LCS of all locally convex spaces with respect to the class Ban of Banach spaces:[8] X▾=EnvBanLCSX. Obviously, X▾ is the inverse limit of the quotient Banach spaces X/U (defined above):

X▾=lim0<-UX/U.
2\. The Stone–Čech compactification β:X->βX of a Tikhonov topological space X is an envelope of X in the category Tikh of all Tikhonov spaces in the class Com of compact spaces with respect to the same class Com:[8] βX=EnvComComX.
3\. The Arens-Michael envelope[9][10][11][12] AAM of a locally convex topological algebra A with a separately continuous multiplication is an envelope of A 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: AAM=EnvBanTopAlgA. The algebra AAM is the inverse limit of the quotient Banach algebras A/U (defined above):

AAM=lim0<-UA/U.
4\. The holomorphic envelope[13] Env𝒪A of a stereotype algebra A is an envelope of A in the category SteAlg of all stereotype algebras in the class DEpi of all dense epimorphisms[14] in SteAlg with respect to the class Ban of all Banach algebras: Env𝒪A=EnvBanDEpiA.
5\. The smooth envelope[15] EnvℰA of a stereotype algebra A is an envelope of A in the category InvSteAlg of all involutive stereotype algebras in the class DEpi of all dense epimorphisms[14] in InvSteAlg with respect to the class DiffMor of all differential homomorphisms into various C*-algebras with joined self-adjoined nilpotent elements: EnvℰA=EnvDiffMorDEpiA.
6\. The continuous envelope[16][17] Env𝒞A of a stereotype algebra A is an envelope of A in the category InvSteAlg of all involutive stereotype algebras in the class DEpi of all dense epimorphisms[14] in InvSteAlg with respect to the class C* of all C*-algebras: Env𝒞A=EnvC*DEpiA.

## Applications

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

* the Gelfand transform GA:A->C(SpecA) of a commutative involutive stereotype algebra A is a continuous envelope of A;[18][19]

* for each locally compact abelian group G the Fourier transform FA:C⋆(G)->C(G^) is a continuous envelope of the stereotype group algebra C⋆(G) of measures with compact support on G.[18]

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).[21][18][20][22]

## See also

* Refinement

## Notes

1. ↑ Akbarov 2016, p. 42.
2. ↑ Cod⁡φ means the codomain of the morphism φ.
3. ↑ Akbarov 2016, Theorem 3.37.
4. ↑ Akbarov 2016, Theorem 3.38.
5. ↑ A category K is said to be co-well-powered in a class of morphisms Ω, if for each object X the category ΩX of all morphisms in Ω going from X is skeletally small.
6. ↑ Akbarov 2016, Theorem 3.60.
7. ↑ A category K is said to be co-well-powered in the class of epimorphisms Epi, if for each object X the category EpiX of all morphisms in Epi going from X is skeletally small.
8. ↑ 8.0 8.1 Akbarov 2016, p. 50.
9. ↑ Helemskii 1993, p. 264.
10. ↑ Pirkovskii 2008.
11. ↑ Akbarov 2009, p. 542.
12. ↑ Akbarov 2010, p. 275.
13. ↑ Akbarov 2016, p. 170.
14. ↑ 14.0 14.1 14.2 A morphism (i.e. a continuous unital homomorphism) of stereotype algebras φ:A->B is called dense if its set of values φ(A) is dense in B.
15. ↑ Akbarov 2017b, p. 741.
16. ↑ Akbarov 2016, p. 179.
17. ↑ Akbarov 2017b, p. 673.
18. ↑ 18.0 18.1 18.2 Akbarov 2016.
19. ↑ Akbarov 2013.
20. ↑ 20.0 20.1 Akbarov 2017b.
21. ↑ Akbarov 2009.
22. ↑ Kuznetsova 2013.

## References

* Helemskii, A.Ya. (1993). Banach and locally convex algebras. Oxford Science Publications. Clarendon Press.
* Pirkovskii, A.Yu. (2008). "Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras". Trans. Moscow Math. Soc. 69: 27–104. doi:10.1090/S0077-1554-08-00169-6. https://www.ams.org/journals/mosc/2008-69-00/S0077-1554-08-00169-6/S0077-1554-08-00169-6.pdf.
* Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences 162 (4): 459–586. doi:10.1007/s10958-009-9646-1.
* Akbarov, S.S. (2010). Stereotype algebras and duality for Stein groups (Thesis). Moscow State University.
* Akbarov, S.S. (2016). "Envelopes and refinements in categories, with applications to functional analysis". Dissertationes Mathematicae 513: 1–188. doi:10.4064/dm702-12-2015. https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513.
* Akbarov, S.S. (2017a). "Continuous and smooth envelopes of topological algebras. Part 1". Journal of Mathematical Sciences 227 (5): 531–668. doi:10.1007/s10958-017-3599-6.
* Akbarov, S.S. (2017b). "Continuous and smooth envelopes of topological algebras. Part 2". Journal of Mathematical Sciences 227 (6): 669–789. doi:10.1007/s10958-017-3600-4.
* Akbarov, S.S. (2013). "The Gelfand transform as a C*-envelope". Mathematical Notes 94 (5–6): 814–815. doi:10.1134/S000143461311014X.
* Kuznetsova, Y. (2013). "A duality for Moore groups". Journal of Operator Theory 69 (2): 101–130. doi:10.7900/jot.2011mar17.1920. Bibcode: 2009arXiv0907.1409K.

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