We follow the definition of stacks project. So the existence of zero object is not in the definition of preadditive categories.
Clearly any ring $R$ (commutative with 1) thought of as a category with only one object, is a preadditive category, where $\hom(R,R):=\hom_{R-mod}(R,R)\cong R$. It is not additive because the zero object doesn't exist.
Is there any other examples of preadditive categories that are not additive? Preferably with a lot of objects and some normal algebra structures, like modules, sheaves, rings, vector spaces, etc.
Best Answer
I don't think this is really any more satisfying than the example of a one-object category, but you can start with your favorite additive category and then delete objects so that biproducts don't exist. For example, you can consider the category of vector spaces of dimension $\le d$ for some fixed positive integer $d$.
The distinction between additive and pre-additive categories is really not that important, for the following reason. Every pre-additive category $C$ admits an additive completion $\widetilde{C}$ in which one formally adjoins the zero object and finite biproducts, and this additive completion is unique in a very strong sense: morphisms between any two objects in $\widetilde{C}$ are completely determined as matrices of morphisms between their direct summands in $C$, so there's no choice whatsoever about how to adjoin the zero object and finite biproducts. Moreover, the inclusion $C \to \widetilde{C}$ preserves any biproducts which already exist in $C$ (note that the analogous statement for, say, the Yoneda embedding is quite false). $\widetilde{C}$ also has the left adjoint property that the category of functors from $C$ to any additive category $D$ is equivalent to the category of functors from $\widetilde{C}$ to $D$, so e.g. their $\text{Ab}$-valued presheaf categories are equivalent.
Applying this construction to the one-object pre-additive category with endomorphisms a ring $R$ produces the category of finite free $R$-modules. So the difference between these two categories is very mild. Similarly, applying it to the category of finite free $R$-modules of rank $\le d$ also produces the category of finite free $R$-modules.
Among other things, this shows that every example of a pre-additive category is obtained from an additive category by deleting some objects.