We delve into the concept of categories with products that distribute over coproducts, which we call doubly-infinitary distributive categories. We show various instances of doubly-infinitary distributive categories aiming for a comparative analysis with established notions such as extensivity, infinitary distributiveness, and cartesian closedness. Our exploration reveals that this condition represents a substantial extension beyond the classical understanding of infinitary distributive categories. Our main theorem establishes that free doubly-infinitary distributive categories are cartesian closed. We end the paper with remarks on non-canonical isomorphisms, open questions, and future work.

翻译：我们深入探讨了乘积在余乘积上具有分配性的范畴概念，称之为双无限分配范畴。我们展示了双无限分配范畴的若干实例，旨在与既有的概念（如可扩展性、无限分配性及笛卡尔闭性）进行比较分析。研究揭示，这一条件代表了对经典无限分配范畴理解的实质性扩展。我们的主要定理证明，自由双无限分配范畴是笛卡尔闭的。文章最后对非标准同构、开放问题及未来研究方向进行了评述。