In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called envelope.
Definition Suppose K {\displaystyle K} is a category, X {\displaystyle X} an object in K {\displaystyle K} , and Γ {\displaystyle \Gamma } and Φ {\displaystyle \Phi } two classes of morphisms in K {\displaystyle K} . The definition of a refinement of X {\displaystyle X} in the class Γ {\displaystyle \Gamma } by means of 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 enrichment of the object X {\displaystyle X} in the class of morphisms Γ {\displaystyle \Gamma } by means of the class of morphisms Φ {\displaystyle \Phi } , if σ ∈ Γ {\displaystyle \sigma \in \Gamma } , and for any morphism φ : B → X {\displaystyle \varphi :B\to X} from the class Φ {\displaystyle \Phi } there exists a unique morphism φ ′ : B → X ′ {\displaystyle \varphi ':B\to X'} in K {\displaystyle K} such that φ = σ ∘ φ ′ {\displaystyle \varphi =\sigma \circ \varphi '} .
An enrichment ρ : E → X {\displaystyle \rho :E\to X} of the object X {\displaystyle X} in the class of morphisms Γ {\displaystyle \Gamma } by means of the class of morphisms Φ {\displaystyle \Phi } is called a refinement of X {\displaystyle X} in Γ {\displaystyle \Gamma } by means of Φ {\displaystyle \Phi } , if for any other enrichment σ : X ′ → X {\displaystyle \sigma :X'\to X} (of X {\displaystyle X} in Γ {\displaystyle \Gamma } by means of Φ {\displaystyle \Phi } ) there is a unique morphism υ : E → X ′ {\displaystyle \upsilon :E\to X'} in K {\displaystyle K} such that ρ = σ ∘ υ {\displaystyle \rho =\sigma \circ \upsilon } . The object E {\displaystyle E} is also called a refinement of X {\displaystyle X} in Γ {\displaystyle \Gamma } by means of Φ {\displaystyle \Phi } . Notations:
ρ = ref Φ Γ X , E = Ref Φ Γ X . {\displaystyle \rho =\operatorname {ref} _{\Phi }^{\Gamma }X,\qquad E=\operatorname {Ref} _{\Phi }^{\Gamma }X.}
In a special case when Γ {\displaystyle \Gamma } 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 Γ {\displaystyle \Gamma } with L {\displaystyle L} in the notations (and in the terms):
ρ = ref Φ L X , E = Ref Φ L X . {\displaystyle \rho =\operatorname {ref} _{\Phi }^{L}X,\qquad E=\operatorname {Ref} _{\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 = ref M Γ X , E = Ref M Γ X . {\displaystyle \rho =\operatorname {ref} _{M}^{\Gamma }X,\qquad E=\operatorname {Ref} _{M}^{\Gamma }X.}
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