什么时候乘积同构于余积
When is the Product isomorphic to the Coproduct
摘要：
对于范畴C，我们研究了当从C^I到C的余积$\bigoplus$和积函子$\prod$对于一个固定集合I是同构的，或者等价地，当这两个函子是Frobenius函子时的问题。我们证明，对于AB-范畴C，这发生当且仅当集合I是有限的。此外，即使在更一般的情况下也是如此，如果C中存在一个态射关于态射的加法是可逆的。如果C不具有这个性质，我们给出一个例子，看看这两个函子可以同构的无限集$I$。然而，我们证明了$\bigoplus$和$\prod$在C ^I的一个合适的子范畴上总是同构的，这个子范畴同构于C^I但不是一个满子范畴.对于模范畴的情况下，我们提供了一个不同的证明，以显示一个有趣的连接的概念Frobenius corings。
英文摘要：
For a category C we investigate the problem of when the coproduct $\bigoplus$ and the product functor $\prod$ from C^I to C are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab-category C this happens if and only if the set I is finite. Moreover, this is true even in a much more general case, if there is a morphism in C that is invertible with respect to the addition of morphisms. If C does not have this property we give an example to see that the two functors can be isomorphic for infinite sets $I$. However we show that $\bigoplus$ and $\prod$ are always isomorphic on a suitable subcategory of C^I which is isomorphic to C^I but is not a full subcategory. For the module category case we provide a different proof to display an interesting connection to the notion of Frobenius corings.
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