envelope (category theory)

Suppose $K$ is a category, $X$ an object in $K$, and $\backslash Omega$ and $\backslash Phi$ two classes of morphisms in $K$. The definition{{sfn|Akbarov|2016|p=42}} of an envelope of $X$ in the class $\backslash Omega$ with respect to the class $\backslash Phi$ consists of two steps.

File:Extension-3.jpg

• A morphism $\backslash sigma:X\backslash to\; X\text{'}$ in $K$ is called an extension of the object $X$ in the class of morphisms $\backslash Omega$ with respect to the class of morphisms $\backslash Phi$, if $\backslash sigma\backslash in\backslash Omega$, and for any morphism $\backslash varphi:X\backslash to\; B$ from the class $\backslash Phi$ there exists a unique morphism $\backslash varphi\text{'}:X\text{'}\backslash to\; B$ in $K$ such that $\backslash varphi=\backslash varphi\text{'}\backslash circ\backslash sigma$.

File:Envelope-3.jpg

• An extension $\backslash rho:X\backslash to\; E$ of the object $X$ in the class of morphisms $\backslash Omega$ with respect to the class of morphisms $\backslash Phi$ is called an envelope of $X$ in $\backslash Omega$ with respect to $\backslash Phi$, if for any other extension $\backslash sigma:X\backslash to\; X\text{'}$ (of $X$ in $\backslash Omega$ with respect to $\backslash Phi$) there is a unique morphism $\backslash upsilon:X\text{'}\backslash to\; E$ in $K$ such that $\backslash rho=\backslash upsilon\backslash circ\backslash sigma$. The object $E$ is also called an envelope of $X$ in $\backslash Omega$ with respect to $\backslash Phi$.

Notations:

\rho=\text{env}_{\Phi}^{\Omega}X, \qquad

E=\text{Env}_{\Phi}^{\Omega}X.

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

\rho=\text{env}_{\Phi}^L X, \qquad

E=\text{Env}_{\Phi}^L X.

Similarly, if $\backslash Phi$ is a class of all morphisms whose ranges belong to a given class of objects $M$ in $K$ it is convenient to replace $\backslash Phi$ 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.

Suppose that to each object $X\backslash in\backslash operatorname\{Ob\}(\{K\})$ in a category $\{K\}$ it is assigned a subset $\{\backslash mathcal\; N\}^X$ in the class

$\backslash operatorname\{Epi\}^X$ of all epimorphisms of the category $\{K\}$, going from $X$, and the following three requirements are fulfilled:

• for each object $X$ the set $\{\backslash mathcal\; N\}^X$ is non-empty and is directed to the left with respect to the pre-order inherited from $\backslash operatorname\{Epi\}^X$

:: $\backslash forall\; \backslash sigma,\backslash sigma\text{'}\backslash in\; \{\backslash mathcal\; N\}^X\backslash quad\; \backslash exists\backslash rho\backslash in\{\backslash mathcal\; N\}^X\backslash quad$

\rho\to\sigma\ \& \ \rho\to\sigma',

• for each object $X$ the covariant system of morphisms generated by $\{\backslash mathcal\; N\}^X$

:: $\backslash \{\backslash iota\_\backslash rho^\backslash sigma;\backslash \; \backslash rho,\backslash sigma\backslash in\{\backslash mathcal\; N\}^X,\backslash \; \backslash rho\backslash to\backslash sigma\backslash \}$

:has a colimit $\backslash varprojlim\; \{\backslash mathcal\; N\}^X$ in $K$, called the local limit in $X$;

• for each morphism $\backslash alpha:X\backslash to\; Y$ and for each element $\backslash tau\backslash in\{\backslash mathcal\; N\}^Y$ there are an element $\backslash sigma\backslash in\{\backslash mathcal\; N\}^X$ and a morphism $\backslash alpha\_\backslash sigma^\backslash tau:\backslash operatorname\{Cod\}\backslash sigma\backslash to\backslash operatorname\{Cod\}\backslash tau$$\backslash operatorname\{Cod\}\backslash varphi$ means the codomain of the morphism $\backslash varphi$. such that

:: $\backslash tau\backslash circ\backslash alpha=\backslash alpha\_\backslash sigma^\backslash tau\backslash circ\backslash sigma.$

Then the family of sets $\{\backslash mathcal\; N\}=\backslash \{\{\backslash mathcal\; N\}^X;\backslash \; X\backslash in\backslash operatorname\{Ob\}(\{K\})\backslash \}$ 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 $U\backslash subseteq\; X$ let us consider its kernel $\backslash operatorname\{Ker\}U=\backslash bigcap\_\{\backslash varepsilon>0\}\backslash varepsilon\backslash cdot\; U$ and the quotient space $X/\backslash operatorname\{Ker\}U$ endowed with the normed topology with the unit ball $U+\backslash operatorname\{Ker\}U$, and let $X/U=(X/\backslash operatorname\{Ker\}U)^\backslash blacktriangledown$ be the completion of $X/\backslash operatorname\{Ker\}U$ (obviously, $X/U$ is a Banach space, and it is called the quotient Banach space of $X$ by $U$). The system of natural mappings $X\backslash to\; X/U$ is a net of epimorphisms in the category $\backslash text\{LCS\}$ of locally convex topological vector spaces.
2. For each locally convex topological algebra $A$ and for each submultiplicative closed convex balanced neighbourhood of zero $U\backslash subseteq\; X$,

:::: $U\backslash cdot\; U\backslash subseteq\; U$,

:: let us again consider its kernel $\backslash operatorname\{Ker\}U=\backslash bigcap\_\{\backslash varepsilon>0\}\backslash varepsilon\backslash cdot\; U$ and the quotient algebra $A/\backslash operatorname\{Ker\}U$ endowed with the normed topology with the unit ball $U+\backslash operatorname\{Ker\}U$, and let $A/U=(A/\backslash operatorname\{Ker\}U)^\backslash blacktriangledown$ be the completion of $A/\backslash operatorname\{Ker\}U$ (obviously, $A/U$ is a Banach algebra, and it is called the quotient Banach algebra of $X$ by $U$). The system of natural mappings $A\backslash to\; A/U$ is a net of epimorphisms in the category $\backslash text\{LCS\}$ of locally convex topological algebras.

Theorem.{{sfn|Akbarov|2016|loc=Theorem 3.37}} Let $\{\backslash mathcal\; N\}$ be a net of epimorphisms in a category $\{K\}$ that generates a class of morphisms $\backslash varPhi$ on the inside:

:: $\{\backslash mathcal\; N\}\; \backslash subseteq\; \backslash varPhi\; \backslash subseteq\; \backslash operatorname\{Mor\}(\{K\})\; \backslash circ\; \{\backslash mathcal\; N\}.$

Then for any class of epimorphisms $\backslash varOmega$ in $K$, which contains all local limits $\backslash varprojlim\; \{\backslash mathcal\; N\}^X$,

:: $\backslash \{\backslash varprojlim\; \{\backslash mathcal\; N\}^X;\; \backslash \; X\backslash in\backslash operatorname\{Ob\}(K)\backslash \}\backslash subseteq\backslash varOmega\backslash subseteq\backslash operatorname\{Epi\}(K),$

the following holds:

:(i) for each object $X$ in $\{K\}$ the local limit $\backslash varprojlim\; \{\backslash mathcal\; N\}^X$ is an envelope $\backslash operatorname\{env\}\_\backslash varPhi^\backslash varOmega\; X$ in $\backslash varOmega$ with respect to $\backslash varPhi$:

:: $\backslash varprojlim\; \{\backslash mathcal\; N\}^X=\backslash operatorname\{env\}\_\backslash varPhi^\backslash varOmega\; X,$

:(ii) the envelope $\backslash operatorname\{Env\}\_\backslash varPhi^\backslash varOmega$ can be defined as a functor.

Theorem.{{sfn|Akbarov|2016|loc=Theorem 3.38}} Let $\{\backslash mathcal\; N\}$ be a net of epimorphisms in a category $\{K\}$ that generates a class of morphisms $\backslash varPhi$ on the inside:

:: $\{\backslash mathcal\; N\}\; \backslash subseteq\; \backslash varPhi\; \backslash subseteq\; \backslash operatorname\{Mor\}(\{K\})\; \backslash circ\; \{\backslash mathcal\; N\}.$

Then for any monomorphically complementable class of epimorphisms $\backslash varOmega$ in $K$ such that $K$ is co-well-poweredA category $K$ is said to be co-well-powered in a class of morphisms $\backslash varOmega$, if for each object $X$ the category $\backslash varOmega^X$ of all morphisms in $\backslash varOmega$ going from $X$ is skeletally small. in $\backslash varOmega$ the envelope $\backslash operatorname\{Env\}\_\backslash varPhi^\backslash varOmega$ can be defined as a functor.

Theorem.{{sfn|Akbarov|2016|loc=Theorem 3.60}}

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 $\backslash operatorname\{Epi\}$,A category $K$ is said to be co-well-powered in the class of epimorphisms $\backslash operatorname\{Epi\}$, if for each object $X$ the category $\backslash operatorname\{Epi\}^X$ of all morphisms in $\backslash operatorname\{Epi\}$ going from $X$ is skeletally small.

:(iv) $\backslash operatorname\{Mor\}(K,L)$ goes from $K$:

:: ,

:(v) $L$ differs morphisms on the outside: for any two different parallel morphisms $\backslash alpha\backslash ne\backslash beta:X\backslash to\; Y$ there is a morphism $\backslash varphi:Y\backslash to\; Z\backslash in\; L$ such that $\backslash varphi\backslash circ\backslash alpha\backslash ne\backslash varphi\backslash circ\backslash beta$,

:(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 $\backslash operatorname\{Cod\}\backslash alpha\backslash in\; L$, then $\backslash operatorname\{Im\}\_\backslash infty\backslash alpha\backslash in\; L$.

Then the envelope $\backslash operatorname\{Env\}\_L^L$ can be defined as a functor.

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

: 1. The completion $X^\backslash blacktriangledown$ of a locally convex topological vector space $X$ is an envelope of $X$ in the category $\backslash text\{LCS\}$ of all locally convex spaces with respect to the class $\backslash text\{Ban\}$ of Banach spaces:{{sfn|Akbarov|2016|p=50}} $X^\backslash blacktriangledown=\backslash text\{Env\}\_\{\backslash text\{Ban\}\}^\{\backslash text\{LCS\}\}X$. Obviously, $X^\backslash blacktriangledown$ is the inverse limit of the quotient Banach spaces $X/U$ (defined above):

::::$X^\backslash blacktriangledown=\backslash lim\_\{0\backslash gets\; U\}X/U.$

: 2. The Stone–Čech compactification $\backslash beta:X\backslash to\backslash beta\; X$ of a Tikhonov topological space $X$ is an envelope of $X$ in the category $\backslash text\{Tikh\}$ of all Tikhonov spaces in the class $\backslash text\{Com\}$ of compact spaces with respect to the same class $\backslash text\{Com\}$:{{sfn|Akbarov|2016|p=50}} $\backslash beta\; X=\backslash text\{Env\}\_\{\backslash text\{Com\}\}^\{\backslash text\{Com\}\}X.$

: 3. The Arens-Michael envelope{{sfn|Helemskii|1993|p=264}}{{sfn|Pirkovskii|2008}}{{sfn|Akbarov|2009|p=542}}{{sfn|Akbarov|2010|p=275}} $A^\{\backslash text\{AM\}\}$ of a locally convex topological algebra $A$ with a separately continuous multiplication is an envelope of $A$ in the category $\backslash text\{TopAlg\}$ of all (locally convex) topological algebras (with separately continuous multiplications) in the class $\backslash text\{TopAlg\}$ with respect to the class $\backslash text\{Ban\}$ of Banach algebras: $A^\{\backslash text\{AM\}\}=\; \backslash text\{Env\}\_\{\backslash text\{Ban\}\}^\{\backslash text\{TopAlg\}\}A$. The algebra $A^\{\backslash text\{AM\}\}$ is the inverse limit of the quotient Banach algebras $A/U$ (defined above):

::::$A^\{\backslash text\{AM\}\}=\backslash lim\_\{0\backslash gets\; U\}A/U.$

: 4. The holomorphic envelope{{sfn|Akbarov|2016|p=170}} $\backslash text\{Env\}\_\{\backslash cal\; O\}\; A$ of a stereotype algebra $A$ is an envelope of $A$ in the category $\backslash text\{SteAlg\}$ of all stereotype algebras in the class $\backslash text\{DEpi\}$ of all dense epimorphismsA morphism (i.e. a continuous unital homomorphism) of stereotype algebras $\backslash varphi:A\backslash to\; B$ is called dense if its set of values $\backslash varphi(A)$ is dense in $B$. in $\backslash text\{SteAlg\}$ with respect to the class $\backslash text\{Ban\}$ of all Banach algebras: $\backslash text\{Env\}\_\{\backslash cal\; O\}\; A=\; \backslash text\{Env\}\_\{\backslash text\{Ban\}\}^\{\backslash text\{DEpi\}\}A.$

: 5. The smooth envelope{{sfn|Akbarov|2017b|p=741}} $\backslash text\{Env\}\_\{\backslash cal\; E\}\; A$ of a stereotype algebra $A$ is an envelope of $A$ in the category $\backslash text\{InvSteAlg\}$ of all involutive stereotype algebras in the class $\backslash text\{DEpi\}$ of all dense epimorphisms in $\backslash text\{InvSteAlg\}$ with respect to the class $\backslash text\{DiffMor\}$ of all differential homomorphisms into various C*-algebras with joined self-adjoined nilpotent elements: $\backslash text\{Env\}\_\{\backslash cal\; E\}\; A=\; \backslash text\{Env\}\_\{\backslash text\{DiffMor\}\}^\{\backslash text\{DEpi\}\}A.$

: 6. The continuous envelope{{sfn|Akbarov|2016|p=179}}{{sfn|Akbarov|2017b|p=673}} $\backslash text\{Env\}\_\{\backslash cal\; C\}\; A$ of a stereotype algebra $A$ is an envelope of $A$ in the category $\backslash text\{InvSteAlg\}$ of all involutive stereotype algebras in the class $\backslash text\{DEpi\}$ of all dense epimorphisms in $\backslash text\{InvSteAlg\}$ with respect to the class $\backslash text\{C\}^*$ of all C*-algebras: $\backslash text\{Env\}\_\{\backslash cal\; C\}\; A=\; \backslash text\{Env\}\_\{\backslash text\{C\}^*\}^\{\backslash text\{DEpi\}\}A.$

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

• the Gelfand transform $G\_A:A\backslash to\; C(\backslash text\{Spec\}\; A)$ of a commutative involutive stereotype algebra $A$ is a continuous envelope of $A$;{{sfn|Akbarov|2016}}{{sfn|Akbarov|2013}}

• for each locally compact abelian group $G$ the Fourier transform $F\_A:C^\backslash star(G)\backslash to\; C(\backslash widehat\{G\})$ is a continuous envelope of the stereotype group algebra $C^\backslash star(G)$ of measures with compact support on $G$.{{sfn|Akbarov|2016}}

In abstract harmonic analysis the notion of envelope plays a key role in the generalizations of the Pontryagin duality theory{{sfn|Akbarov|2017b}} 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).{{sfn|Akbarov|2009}}{{sfn|Akbarov|2016}}{{sfn|Akbarov|2017b}}{{sfn|Kuznetsova|2013}}

• Refinement

{{reflist}}

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• {{cite journal|last=Akbarov|first=S.S.|title=The Gelfand transform as a C*-envelope |journal=Mathematical Notes|year=2013|volume=94|issue=5–6|pages= 814–815|doi=10.1134/S000143461311014X|s2cid=121354607}}
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{{refend}}

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