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 {\displaystyle K} is a category, X {\displaystyle X} an object in K {\displaystyle K}, and Ω {\displaystyle \Omega } and Φ {\displaystyle \Phi } two classes of morphisms in K {\displaystyle K}. The definition of an envelope of X {\displaystyle X} in the class Ω {\displaystyle \Omega } with respect to the class Φ {\displaystyle \Phi } consists of two steps.

A morphism σ : X → X ′ {\displaystyle \sigma :X\to X'} in K {\displaystyle K} is called an extension of the object X {\displaystyle X} in the class of morphisms Ω {\displaystyle \Omega } with respect to the class of morphisms Φ {\displaystyle \Phi }, if σ ∈ Ω {\displaystyle \sigma \in \Omega }, and for any morphism φ : X → B {\displaystyle \varphi :X\to B} from the class Φ {\displaystyle \Phi } there exists a unique morphism φ ′ : X ′ → B {\displaystyle \varphi ':X'\to B} in K {\displaystyle K} such that φ = φ ′ ∘ σ {\displaystyle \varphi =\varphi '\circ \sigma }.

An extension ρ : X → E {\displaystyle \rho :X\to E} of the object X {\displaystyle X} in the class of morphisms Ω {\displaystyle \Omega } with respect to the class of morphisms Φ {\displaystyle \Phi } is called an envelope of X {\displaystyle X} in Ω {\displaystyle \Omega } with respect to Φ {\displaystyle \Phi }, if for any other extension σ : X → X ′ {\displaystyle \sigma :X\to X'} (of X {\displaystyle X} in Ω {\displaystyle \Omega } with respect to Φ {\displaystyle \Phi }) there is a unique morphism υ : X ′ → E {\displaystyle \upsilon :X'\to E} in K {\displaystyle K} such that ρ = υ ∘ σ {\displaystyle \rho =\upsilon \circ \sigma }. The object E {\displaystyle E} is also called an envelope of X {\displaystyle X} in Ω {\displaystyle \Omega } with respect to Φ {\displaystyle \Phi }.

Notations:

ρ = env Φ Ω X , E = Env Φ Ω X . {\displaystyle \rho ={\text{env}}_{\Phi }^{\Omega }X,\qquad E={\text{Env}}_{\Phi }^{\Omega }X.}

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

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

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

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

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

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

Nets of epimorphisms and functoriality

Suppose that to each object X ∈ Ob ⁡ ( K ) {\displaystyle X\in \operatorname {Ob} ({K})} in a category K {\displaystyle {K}} it is assigned a subset N X {\displaystyle {\mathcal {N}}^{X}} in the class Epi X {\displaystyle \operatorname {Epi} ^{X}} of all epimorphisms of the category K {\displaystyle {K}}, going from X {\displaystyle X}, and the following three requirements are fulfilled:

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

∀ σ , σ ′ ∈ N X ∃ ρ ∈ N X ρ → σ & ρ → σ ′ , {\displaystyle \forall \sigma ,\sigma '\in {\mathcal {N}}^{X}\quad \exists \rho \in {\mathcal {N}}^{X}\quad \rho \to \sigma \ \&\ \rho \to \sigma ',}

for each object X {\displaystyle X} the covariant system of morphisms generated by N X {\displaystyle {\mathcal {N}}^{X}}

{ ι ρ σ ; ρ , σ ∈ N X , ρ → σ } {\displaystyle \{\iota _{\rho }^{\sigma };\ \rho ,\sigma \in {\mathcal {N}}^{X},\ \rho \to \sigma \}}

has a colimit lim ← ⁡ N X {\displaystyle \varprojlim {\mathcal {N}}^{X}} in K {\displaystyle K}, called the local limit in X {\displaystyle X};

for each morphism α : X → Y {\displaystyle \alpha :X\to Y} and for each element τ ∈ N Y {\displaystyle \tau \in {\mathcal {N}}^{Y}} there are an element σ ∈ N X {\displaystyle \sigma \in {\mathcal {N}}^{X}} and a morphism α σ τ : Cod ⁡ σ → Cod ⁡ τ {\displaystyle \alpha _{\sigma }^{\tau }:\operatorname {Cod} \sigma \to \operatorname {Cod} \tau } such that

τ ∘ α = α σ τ ∘ σ . {\displaystyle \tau \circ \alpha =\alpha _{\sigma }^{\tau }\circ \sigma .}

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

Examples.

For each locally convex topological vector space X {\displaystyle X} and for each closed convex balanced neighbourhood of zero U ⊆ X {\displaystyle U\subseteq X} let us consider its kernel Ker ⁡ U = ⋂ ε > 0 ε ⋅ U {\displaystyle \operatorname {Ker} U=\bigcap _{\varepsilon >0}\varepsilon \cdot U} and the quotient space X / Ker ⁡ U {\displaystyle X/\operatorname {Ker} U} endowed with the normed topology with the unit ball U + Ker ⁡ U {\displaystyle U+\operatorname {Ker} U}, and let X / U = ( X / Ker ⁡ U ) ▾ {\displaystyle X/U=(X/\operatorname {Ker} U)^{\blacktriangledown }} be the completion of X / Ker ⁡ U {\displaystyle X/\operatorname {Ker} U} (obviously, X / U {\displaystyle X/U} is a Banach space, and it is called the quotient Banach space of X {\displaystyle X} by U {\displaystyle U}). The system of natural mappings X → X / U {\displaystyle X\to X/U} is a net of epimorphisms in the category LCS {\displaystyle {\text{LCS}}} of locally convex topological vector spaces.
For each locally convex topological algebra A {\displaystyle A} and for each submultiplicative closed convex balanced neighbourhood of zero U ⊆ X {\displaystyle U\subseteq X},

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

Theorem. Let N {\displaystyle {\mathcal {N}}} be a net of epimorphisms in a category K {\displaystyle {K}} that generates a class of morphisms Φ {\displaystyle \varPhi } on the inside:

N ⊆ Φ ⊆ Mor ⁡ ( K ) ∘ N . {\displaystyle {\mathcal {N}}\subseteq \varPhi \subseteq \operatorname {Mor} ({K})\circ {\mathcal {N}}.}

Then for any class of epimorphisms Ω {\displaystyle \varOmega } in K {\displaystyle K}, which contains all local limits lim ← ⁡ N X {\displaystyle \varprojlim {\mathcal {N}}^{X}},

{ lim ← ⁡ N X ; X ∈ Ob ⁡ ( K ) } ⊆ Ω ⊆ Epi ⁡ ( K ) , {\displaystyle \{\varprojlim {\mathcal {N}}^{X};\ X\in \operatorname {Ob} (K)\}\subseteq \varOmega \subseteq \operatorname {Epi} (K),}

the following holds:

(i) for each object X {\displaystyle X} in K {\displaystyle {K}} the local limit lim ← ⁡ N X {\displaystyle \varprojlim {\mathcal {N}}^{X}} is an envelope env Φ Ω ⁡ X {\displaystyle \operatorname {env} _{\varPhi }^{\varOmega }X} in Ω {\displaystyle \varOmega } with respect to Φ {\displaystyle \varPhi }: lim ← ⁡ N X = env Φ Ω ⁡ X , {\displaystyle \varprojlim {\mathcal {N}}^{X}=\operatorname {env} _{\varPhi }^{\varOmega }X,}

(ii) the envelope Env Φ Ω {\displaystyle \operatorname {Env} _{\varPhi }^{\varOmega }} can be defined as a functor.

Theorem. Let N {\displaystyle {\mathcal {N}}} be a net of epimorphisms in a category K {\displaystyle {K}} that generates a class of morphisms Φ {\displaystyle \varPhi } on the inside:

N ⊆ Φ ⊆ Mor ⁡ ( K ) ∘ N . {\displaystyle {\mathcal {N}}\subseteq \varPhi \subseteq \operatorname {Mor} ({K})\circ {\mathcal {N}}.}

Then for any monomorphically complementable class of epimorphisms Ω {\displaystyle \varOmega } in K {\displaystyle K} such that K {\displaystyle K} is co-well-powered in Ω {\displaystyle \varOmega } the envelope Env Φ Ω {\displaystyle \operatorname {Env} _{\varPhi }^{\varOmega }} can be defined as a functor.

Theorem. Suppose a category K {\displaystyle K} and a class of objects L {\displaystyle L} have the following properties:

(i) K {\displaystyle K} is cocomplete,

(ii) K {\displaystyle K} has nodal decomposition,

(iii) K {\displaystyle K} is co-well-powered in the class Epi {\displaystyle \operatorname {Epi} },

(iv) Mor ⁡ ( K , L ) {\displaystyle \operatorname {Mor} (K,L)} goes from K {\displaystyle K}: ∀ X ∈ Ob ⁡ ( K ) ∃ φ ∈ Mor ⁡ ( K ) Dom ⁡ φ = X & Cod ⁡ φ ∈ L {\displaystyle \forall X\in \operatorname {Ob} (K)\quad \exists \varphi \in \operatorname {Mor} (K)\quad \operatorname {Dom} \varphi =X\quad \&\quad \operatorname {Cod} \varphi \in L},

(v) L {\displaystyle L} differs morphisms on the outside: for any two different parallel morphisms α ≠ β : X → Y {\displaystyle \alpha \neq \beta :X\to Y} there is a morphism φ : Y → Z ∈ L {\displaystyle \varphi :Y\to Z\in L} such that φ ∘ α ≠ φ ∘ β {\displaystyle \varphi \circ \alpha \neq \varphi \circ \beta },

(vi) L {\displaystyle L} is closed with respect to passage to colimits,

(vii) L {\displaystyle L} is closed with respect to passage from the codomain of a morphism to its nodal image: if Cod ⁡ α ∈ L {\displaystyle \operatorname {Cod} \alpha \in L}, then Im ∞ ⁡ α ∈ L {\displaystyle \operatorname {Im} _{\infty }\alpha \in L}.

Then the envelope Env L L {\displaystyle \operatorname {Env} _{L}^{L}} can be defined as a functor.

Examples

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

1. The completion X ▾ {\displaystyle X^{\blacktriangledown }} of a locally convex topological vector space X {\displaystyle X} is an envelope of X {\displaystyle X} in the category LCS {\displaystyle {\text{LCS}}} of all locally convex spaces with respect to the class Ban {\displaystyle {\text{Ban}}} of Banach spaces: X ▾ = Env Ban LCS X {\displaystyle X^{\blacktriangledown }={\text{Env}}_{\text{Ban}}^{\text{LCS}}X}. Obviously, X ▾ {\displaystyle X^{\blacktriangledown }} is the inverse limit of the quotient Banach spaces X / U {\displaystyle X/U} (defined above): X ▾ = lim 0 ← U X / U . {\displaystyle X^{\blacktriangledown }=\lim _{0\gets U}X/U.}

2. The Stone–Čech compactification β : X → β X {\displaystyle \beta :X\to \beta X} of a Tikhonov topological space X {\displaystyle X} is an envelope of X {\displaystyle X} in the category Tikh {\displaystyle {\text{Tikh}}} of all Tikhonov spaces in the class Com {\displaystyle {\text{Com}}} of compact spaces with respect to the same class Com {\displaystyle {\text{Com}}}: β X = Env Com Com X . {\displaystyle \beta X={\text{Env}}_{\text{Com}}^{\text{Com}}X.}

3. The Arens-Michael envelope A AM {\displaystyle A^{\text{AM}}} of a locally convex topological algebra A {\displaystyle A} with a separately continuous multiplication is an envelope of A {\displaystyle A} in the category TopAlg {\displaystyle {\text{TopAlg}}} of all (locally convex) topological algebras (with separately continuous multiplications) in the class TopAlg {\displaystyle {\text{TopAlg}}} with respect to the class Ban {\displaystyle {\text{Ban}}} of Banach algebras: A AM = Env Ban TopAlg A {\displaystyle A^{\text{AM}}={\text{Env}}_{\text{Ban}}^{\text{TopAlg}}A}. The algebra A AM {\displaystyle A^{\text{AM}}} is the inverse limit of the quotient Banach algebras A / U {\displaystyle A/U} (defined above): A AM = lim 0 ← U A / U . {\displaystyle A^{\text{AM}}=\lim _{0\gets U}A/U.}

4. The holomorphic envelope Env O A {\displaystyle {\text{Env}}_{\cal {O}}A} of a stereotype algebra A {\displaystyle A} is an envelope of A {\displaystyle A} in the category SteAlg {\displaystyle {\text{SteAlg}}} of all stereotype algebras in the class DEpi {\displaystyle {\text{DEpi}}} of all dense epimorphisms in SteAlg {\displaystyle {\text{SteAlg}}} with respect to the class Ban {\displaystyle {\text{Ban}}} of all Banach algebras: Env O A = Env Ban DEpi A . {\displaystyle {\text{Env}}_{\cal {O}}A={\text{Env}}_{\text{Ban}}^{\text{DEpi}}A.}

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

6. The continuous envelope Env C A {\displaystyle {\text{Env}}_{\cal {C}}A} of a stereotype algebra A {\displaystyle A} is an envelope of A {\displaystyle A} in the category InvSteAlg {\displaystyle {\text{InvSteAlg}}} of all involutive stereotype algebras in the class DEpi {\displaystyle {\text{DEpi}}} of all dense epimorphisms in InvSteAlg {\displaystyle {\text{InvSteAlg}}} with respect to the class C ∗ {\displaystyle {\text{C}}^{*}} of all C*-algebras: Env C A = Env C ∗ DEpi A . {\displaystyle {\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 → C ( Spec A ) {\displaystyle G_{A}:A\to C({\text{Spec}}A)} of a commutative involutive stereotype algebra A {\displaystyle A} is a continuous envelope of A {\displaystyle A};

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

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).

Notes