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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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