Is there a convention for precedence of operators in an additive category?

Question

The laws for an additive category are that there must be a zero object, binary products, that every Hom-set is an abelian group, and that the morphism addition distributes over composition.

My question is a question about convention. If someone writes an expression $f + g \circ h$, how should I parse this? As $(f + g) \circ h$ or as $f + (g \circ h)$?

Obviously, conventions are conventions, so if the question comes up, which would have been used by Grothendieck or MacLane. If there is no general agreement, that works too.

Answer 1

I'd read it as $f+(g \circ h)$, but I imagine it'll be clear from context. If there's cause for confusion I'd imagine brackets would usually be inserted.

In most cases, you can work it out by looking at domains and codomains:

• $f+(g \circ h)$ is only defined when $f$ and $g \circ h$ have the same domain and codomain;
• $(f+g) \circ h$ is only defined when $f$ and $g$ have the same domain and codomain.

Answer 2

A linear category is a ringoid (a ring with multiple objects). In a ring we have the usual order of operations. The same convention makes sense for ringoids.