The category of internal coalgebras in a cocomplete category 𝒞 with respect to a variety 𝒱 is equivalent to the category of left adjoint functors from 𝒱 to 𝒞. This can be seen best when considering such coalgebras as finite coproduct preserving functors from 𝒯𝒱op, the dual of the Lawvere theory of 𝒱, into 𝒞: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of 𝒯𝒱op into Alg𝒯. Since SMod-coalgebras in the variety RMod for rings R and S are nothing but left S-, right R-bimodules, the equivalence above generalizes the Eilenberg–Watts theorem and all its previous generalizations. By generalizing and strengthening Bergman’s completeness result for categories of internal coalgebras in varieties, we also prove that the category of coalgebras in a locally presentable category 𝒞 is locally presentable and comonadic over 𝒞 and, hence, complete in particular. We show, moreover, that Freyd’s canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where 𝒱 is a commutative variety, are coreflectors from the category Coalg(𝒯 ,𝒱) into 𝒱.

中文翻译：

---

共完备范畴中的内部余代数：推广 Eilenberg-Watts 定理

共完备范畴中的内部代数范畴𝒞关于品种𝒱等价于左伴随函子的范畴𝒱到𝒞. 当将这样的余代数视为有限余积保持函子时，可以最好地看到这一点𝒯𝒱○p，劳维尔理论的对偶𝒱， 进入𝒞：余代数是左伴随的限制，任何这样的左伴随都是余代数沿嵌入的左 Kan 扩展𝒯𝒱○p进入一种lG𝒯. 自从小号模组- 各种代数R模组用于戒指R和小号什么都没有小号-， 正确的R-bimodules，上面的等价式推广了 Eilenberg-Watts 定理及其所有先前的推广。通过推广和加强Bergman关于变种内部余代数范畴的完备性结果，我们还证明了局部可呈现范畴中的余代数范畴𝒞是局部可观的和共通的𝒞因此，特别完整。此外，我们展示了 Freyd 的各种内部余代数的规范构造定义了左伴随函子。各个右伴随物的特殊实例出现在各种代数上下文中，并且在𝒱是一个交换变体，是来自该范畴的共反射器C○一种lG(𝒯 ,𝒱)进入𝒱.