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Points of affine categories and additivity. (English) Zbl 1088.18005
Let $$\mathcal C$$ be a pointed category with finite limits and finite coproducts. For any object $$B$$ in $$\mathcal C$$, let $${\mathcal P} t({\mathcal C}, B)$$ be the category of pointed objects in the comma category $$({\mathcal C}, B)$$, and let $$F_B: {\mathcal C}\to{\mathcal P} t({\mathcal C}, B)$$ be the functor defined by $$F_B(X)= (B+ X, (1,0), i_1)$$.
In this paper, it is proved that the category $$\mathcal C$$ is additive if and only if it has a class $$S$$ of (adequately defined) generators $$B$$ such that the functors $$F_B: {\mathcal C}\to \mathcal P t({\mathcal C}, B)$$ are equivalences of categories.
##### MSC:
 18C05 Equational categories 18C10 Theories (e.g., algebraic theories), structure, and semantics 18C20 Eilenberg-Moore and Kleisli constructions for monads
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