Let 𝒜 be an abelian category with enough projectives, and let 𝒞 ⊆obj (𝒜) satisfy (i) for every o

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Let 𝒜 be an abelian category with enough projectives, and...

Let $\mathcal{A}$ be an abelian category with enough projectives, and let $\mathcal{C} \subseteq$ obj $(\mathcal{A})$ satisfy (i) for every object $A$ in $\mathcal{A}$, there exists $C \in \mathbb{C}$ and an epimorphism $C \rightarrow A$; (ii) if $C \in C{C}$, then every direct summand of $C$ also lies in $C$. Prove that every projective lies in $\mathfrak{C}$. The dual result also holds.

Let $\mathcal{A}$ be an abelian category with enough projectives, and let $\mathcal{C} \subseteq$ obj $(\mathcal{A})$ satisfy
(i) for every object $A$ in $\mathcal{A}$, there exists $C \in \mathbb{C}$ and an epimorphism $C \rightarrow A$;
(ii) if $C \in C{C}$, then every direct summand of $C$ also lies in $C$.
Prove that every projective lies in $\mathfrak{C}$. The dual result also holds.

Key Concepts

Direct Summands and Closure Properties

A direct summand is an object that, together with another object, forms a direct sum decomposition of a larger object. The property of closure under direct summands indicates that if an object in the collection has a direct summand, then this summand also belongs to the collection. This aspect is crucial when showing that certain properties or classes of objects extend to all projective objects.

Epimorphisms and Their Splitting

An epimorphism is a morphism that is right-cancellable, serving as the categorical analog of a surjective function. In the presence of projective objects, any epimorphism from an object in a designated collection can split, meaning that the object being projected onto can be represented as a direct summand of the source object. This splitting is a key technique for transferring properties from one object to its summands.

Projective Object

A projective object in a category is one that satisfies the lifting property; any morphism from it can be lifted along any epimorphism. This property simplifies many constructions in homological algebra, as it allows one to replace arbitrary objects with projective ones when studying exact sequences and constructing resolutions.

Abelian Category with Enough Projectives

An abelian category is a category in which morphisms and objects can be added and where kernels and cokernels exist and behave well. The property of having enough projectives means that for every object, there is a projective object mapping onto it via an epimorphism. This guarantees the ability to construct projective resolutions and is crucial for many homological algebra techniques.

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