There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{ A}$, the isomorphisms $\operatorname{ Ext}^n (\bigoplus_{ i\in I}A_{ i},X) \cong \prod_{ i\in I} \operatorname{ Ext}^n(A_{ i},X)$ and $\operatorname{ Ext}^n (X,\prod_{ i\in I}A_{ i}) \cong \prod_{ i\in I}\operatorname{ Ext}^n (X,A_{ i})$ always exist. The goal of this paper is to show similar isomorphisms for the Yoneda Ext in Ab4 and Ab4* abelian categories with not necessarily enough projectives nor injectives. The desired isomorphisms are constructed explicitely by using limits and colimits.
Categories: math.CT