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.

Definition

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

Extension.

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 $\varphi =\varphi '\circ \sigma$ .

Envelope.

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:

$\rho ={\text{env}}_{\Phi }^{\Omega }X,\qquad E={\text{Env}}_{\Phi }^{\Omega }X.$

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):

$\rho ={\text{env}}_{\Phi }^{L}X,\qquad E={\text{Env}}_{\Phi }^{L}X.$

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):

$\rho ={\text{env}}_{M}^{\Omega }X,\qquad E={\text{Env}}_{M}^{\Omega }X.$

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

$\rho ={\text{env}}_{M}^{L}X,\qquad E={\text{Env}}_{M}^{L}X.$

Examples

The completion $X^{\blacktriangledown }$ of a locally convex topological vector space $X$ is an envelope of $X$ in the category ${\text{LCS}}$ of all locally convex spaces with respect to the class ${\text{Ban}}$ of Banach spaces:^[2] $X^{\blacktriangledown }={\text{Env}}_{\text{Ban}}^{\text{LCS}}X.$
The Stone–Čech compactification $\beta :X\to \beta X$ of a Tikhonov topological space $X$ is an envelope of $X$ in the category ${\text{Tikh}}$ of all Tikhonov spaces in the class ${\text{Com}}$ of compact spaces with respect to the same class ${\text{Com}}$ :^[2] $\beta X={\text{Env}}_{\text{Com}}^{\text{Com}}X.$
The Arens-Michael envelope^[3]^[4]^[5]^[6] $A^{\text{AM}}$ of a locally convex topological algebra $A$ with a separately continuous multiplication is an envelope of $A$ in the category ${\text{TopAlg}}$ of all (locally convex) topological algebras (with separately continuous multiplications) in the class ${\text{TopAlg}}$ with respect to the class ${\text{Ban}}$ of Banach algebras: $A^{\text{AM}}={\text{Env}}_{\text{Ban}}^{\text{TopAlg}}A.$
The holomorphic envelope^[7] ${\text{Env}}_{\cal {O}}A$ of a stereotype algebra $A$ is an envelope of $A$ in the category ${\text{SteAlg}}$ of all stereotype algebras in the class ${\text{DEpi}}$ of all dense epimorphisms^[8] in ${\text{SteAlg}}$ with respect to the class ${\text{Ban}}$ of all Banach algebras: ${\text{Env}}_{\cal {O}}A={\text{Env}}_{\text{Ban}}^{\text{DEpi}}A.$
The smooth envelope^[9] ${\text{Env}}_{\cal {E}}A$ of a stereotype algebra $A$ is an envelope of $A$ in the category ${\text{InvSteAlg}}$ of all involutive stereotype algebras in the class ${\text{DEpi}}$ of all dense epimorphisms^[8] in ${\text{InvSteAlg}}$ with respect to the class ${\text{DiffMor}}$ of all differential homomorphisms into various C*-algebras with joined self-adjoined nilpotent elements: ${\text{Env}}_{\cal {E}}A={\text{Env}}_{\text{DiffMor}}^{\text{DEpi}}A.$
The continuous envelope^[10]^[11] ${\text{Env}}_{\cal {C}}A$ of a stereotype algebra $A$ is an envelope of $A$ in the category ${\text{InvSteAlg}}$ of all involutive stereotype algebras in the class ${\text{DEpi}}$ of all dense epimorphisms^[8] in ${\text{InvSteAlg}}$ with respect to the class ${\text{C}}^{*}$ of all C*-algebras: ${\text{Env}}_{\cal {C}}A={\text{Env}}_{{\text{C}}^{*}}^{\text{DEpi}}A.$

Applications

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

the Gelfand transform $G_{A}:A\to C({\text{Spec}}A)$ of a commutative involutive stereotype algebra $A$ is a continuous envelope of $A$ ^[12]^[13];

for each locally compact abelian group $G$ the Fourier transform $F_{A}:C^{\star }(G)\to C({\widehat {G}})$ is a continuous envelope of the stereotype group algebra $C^{\star }(G)$ of measures with compact support on $G$ ^[12].

In abstract harmonic analysis the notion of envelope plays a key role in the generalizations of the Pontryagin duality theory^[14] to the clases 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).^[15]^[12]^[14]^[16]

Notes

↑ Akbarov 2016, p. 42.
1 2 Akbarov 2016, p. 50.
↑ Helemskii 1993, p. 264.
↑ Pirkovskii 2008.
↑ Akbarov 2009, p. 542.
↑ Akbarov 2010, p. 275.
↑ Akbarov 2016, p. 170.
1 2 3 A 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 $B$ .
↑ Akbarov 2017, p. 741.
↑ Akbarov 2016, p. 179.
↑ Akbarov 2017, p. 673.
1 2 3 Akbarov 2016.
↑ Akbarov 2013.
1 2 Akbarov 2017.
↑ Akbarov 2009.
↑ 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" (PDF). Trans. Moscow Math. Soc. 69: 27–104.
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. arXiv:0806.3205. 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. arXiv:1110.2013. doi:10.4064/dm702-12-2015.
Akbarov, S.S. (2017). "Continuous and smooth envelopes of topological algebras. Part 1". Journal of Mathematical Sciences. 227 (5): 531–668. arXiv:1303.2424. doi:10.1007/s10958-017-3599-6.
Akbarov, S.S. (2017). "Continuous and smooth envelopes of topological algebras. Part 2". Journal of Mathematical Sciences. 227 (6): 669–789. arXiv:1303.2424. 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. arXiv:0907.1409. Bibcode:2009arXiv0907.1409K. doi:10.7900/jot.2011mar17.1920.

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[FOOTNOTEAkbarov201642-1] Akbarov 2016, p. 42.

[FOOTNOTEAkbarov201650-2] 1 2 Akbarov 2016, p. 50.

[FOOTNOTEHelemskii1993264-3] Helemskii 1993, p. 264.

[FOOTNOTEPirkovskii2008-4] Pirkovskii 2008.

[FOOTNOTEAkbarov2009542-5] Akbarov 2009, p. 542.

[FOOTNOTEAkbarov2010275-6] Akbarov 2010, p. 275.

[FOOTNOTEAkbarov2016170-7] Akbarov 2016, p. 170.

[DEpi-8] 1 2 3 A 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 $B$ .

[FOOTNOTEAkbarov2017741-9] Akbarov 2017, p. 741.

[FOOTNOTEAkbarov2016179-10] Akbarov 2016, p. 179.

[FOOTNOTEAkbarov2017673-11] Akbarov 2017, p. 673.

[FOOTNOTEAkbarov2016-12] 1 2 3 Akbarov 2016.

[FOOTNOTEAkbarov2013-13] Akbarov 2013.

[FOOTNOTEAkbarov2017-14] 1 2 Akbarov 2017.

[FOOTNOTEAkbarov2009-15] Akbarov 2009.

[FOOTNOTEKuznetsova2013-16] Kuznetsova 2013.

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