In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism φ : X → Y {\displaystyle \varphi :X\to Y} is a representation of φ {\displaystyle \varphi } as a product φ = σ ∘ β ∘ π {\displaystyle \varphi =\sigma \circ \beta \circ \pi } , where π {\displaystyle \pi } is a strong epimorphism, β {\displaystyle \beta } a bimorphism, and σ {\displaystyle \sigma } a strong monomorphism.
Uniqueness and notations If it exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions φ = σ ∘ β ∘ π {\displaystyle \varphi =\sigma \circ \beta \circ \pi } and φ = σ ′ ∘ β ′ ∘ π ′ {\displaystyle \varphi =\sigma '\circ \beta '\circ \pi '} there exist isomorphisms η {\displaystyle \eta } and θ {\displaystyle \theta } such that
π ′ = η ∘ π , {\displaystyle \pi '=\eta \circ \pi ,}
β = θ ∘ β ′ ∘ η , {\displaystyle \beta =\theta \circ \beta '\circ \eta ,}
σ ′ = σ ∘ θ . {\displaystyle \sigma '=\sigma \circ \theta .}
This property justifies some special notations for the elements of the nodal decomposition:
π = coim ∞ φ , P = Coim ∞ φ , β = red ∞ φ , σ = im ∞ φ , Q = Im ∞ φ , {\displaystyle {\begin{aligned}&\pi =\operatorname {coim} _{\infty }\varphi ,&&P=\operatorname {Coim} _{\infty }\varphi ,\\&\beta =\operatorname {red} _{\infty }\varphi ,&&\\&\sigma =\operatorname {im} _{\infty }\varphi ,&&Q=\operatorname {Im} _{\infty }\varphi ,\end{aligned}}}
– here coim ∞ φ {\displaystyle \operatorname {coim} _{\infty }\varphi } and Coim ∞ φ {\displaystyle \operatorname {Coim} _{\infty }\varphi } are called the nodal coimage of φ {\displaystyle \varphi } , im ∞ φ {\displaystyle \operatorname {im} _{\infty }\varphi } and Im ∞ φ {\displaystyle \operatorname {Im} _{\infty }\varphi } the nodal image of φ {\displaystyle \varphi } , and red ∞ φ {\displaystyle \operatorname {red} _{\infty }\varphi } the nodal reduced part of φ {\displaystyle \varphi } . In these notations the nodal decomposition takes the form
φ = im ∞ φ ∘ red ∞ φ ∘ coim ∞ φ . {\displaystyle \varphi =\operatorname {im} _{\infty }\varphi \circ \operatorname {red} _{\infty }\varphi \circ \operatorname {coim} _{\infty }\varphi .}
Connection with the basic decomposition in pre-abelian categories In a pre-abelian category K {\displaystyle {\mathcal {K}}} each morphism φ {\displaystyle \varphi } has a standard decomposition
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