Let $C$ be a category with a zero object $0$, small products, and small coproducts. Let $(A_i)_{i \in I}$ be a (possibly infinite) list of objects. There is a canonical map $\amalg_{i \in I} A_i \to \prod_{i \in I} A_i$. If this map is an isomorphism, we denote the product / coproduct by $\oplus_{i \in I} A_i$ and call it the *biproduct* of the $A_i$. If the $A_i$'s are all the same, call this the $I$-fold *bipower* $A^{(I)}$ of $A$. When $I$ is finite, I think I have a pretty good handle on $I$-indexed biproducts. But when $I$ is infinite, I get more confused. Here are some things I know or think I know about this situation:

Some categories have all small biproducts. The main examples I'm aware of are the category of suplattices and the category of locally presentable categories (and left adjoint functors). These can be modified to get examples of categories which have $\kappa$-small biproducts but no larger.

Certain infinite biproducts may exist in an additive category. For example, in $Ab^{\mathbb N}$ (where $\mathbb N$ is regarded as a set), the constant functor at $\mathbb Z$ is the biproduct of the representables.

On the other hand, I thought I knew an argument at some point that in an

*additive*category, any infinite bipower of an object must be zero. I can't seem to reproduce the argument, but I think it was some sort of Eilenberg swindle sort of thing...More generally, I want to say that if an infinite biproduct $\oplus_{i \in I} A_i$ exists in an

*additive*category, then we must be "close" to the situation of (2) above, where $Hom(A_i,A_j) = 0$ for $i \neq j$. I don't know how to formulate this, though, if it's correct in any sense at all.

So I suppose my questions are the following:

**Question 1:** Is it true that in an additive category, any infinite bipower which exists must be zero?

**Question 2:** In an additive category, if an infinite biproduct $\oplus_{i \in I} A_i$ exists, is there sense in which it must be "close" to the setting of (2) above? For example, must it be the case that for each $i \in I$, $Hom(A_i,A_j) = 0$ for all but finitely many $j \in I$?