There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\] , given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\] , an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.

中文翻译：

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Abelian 类别中的 YONEDA EXT 和任意副产品

众所周知的恒等式涉及具有足够射影的 AB4 阿贝尔范畴中的 Ext 双函子、联积和积。即，对于每个这样的类别 \[\数学{A}\] ，给定一个对象X和一组对象 \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\] , 同构 \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{ \text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\文本{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] 可以建，在哪里 \[前{t^{\text{n}}}\] 是个nHom 函子的第一个派生函子。本文的目的是展示类似的同构nth Yoneda Ext，它是一个相当于 \[前{t^{\text{n}}}\] 可以在更一般的上下文中定义。所需的同构是通过使用 AB4 阿贝尔范畴中的共界来明确构造的，不一定有足够的射影或射影，扩展了 Colpi 和 Fuller 在 [8] 中的结果。此外，构建的同构用于表征 AB4 类别。还说明了双重结果。