Monoidal operations on categories where the maps on Aut, End are injective

Question

Suppose we have a monoidal category $(\mathcal{C},\otimes)$. I am interested in the conditions or situations where the following do and do not hold:

1. For any objects A, B of $\mathcal{C}$, the induced map $\operatorname{End}(A) \times \operatorname{End}(B) \to \operatorname{End}(A \otimes B)$ is injective (by induced map, I mean the map induced by the bifunctoriality).

2. For any objects A, B of $\mathcal{C}$, the induced map $\operatorname{Aut}(A) \times \operatorname{Aut}(B) \to \operatorname{Aut}(A \otimes B)$ is injective (this is a restriction of the map in (1)).

(1) is stronger than (2). I think I have a proof that in the case that $\otimes$ is a product or coproduct (i.e., we have a Cartesian or co-Cartesian (?) monoidal category) [EDIT: My proof was wrong, as the counterexamples below show], both (1) and (2) hold, i.e., the maps on the two factors are "determined" by the map on the (co)-product. On the other hand, when $\otimes$ is the tensor product of modules over a commutative unital ring, (1) and (2) need not hold, though I think they do hold when the ring is a field.

So my question is: what are more examples where (1) and (2) do hold, what are more examples where they don't, and are there some other conditions/properties which would imply (1) and (2)?

ADDED LATER: It seems that zero objects provide some immediate counterexamples to (1) and (2) even for the category of sets. So I'm modifying (1) and (2) to the following:

1.' For what objects A, B, is the induced map $\operatorname{End}(A) \times \operatorname{End}(B) \to \operatorname{End}(A \otimes B)$ injective? i.e., How would we characterize such A and B? The most general possibility seems to be if it is true for all A and B that are not initial or final objects.

2.' For what objects A, B, is the induced map $\operatorname{Aut}(A) \times \operatorname{Aut}(B) \to \operatorname{Aut}(A \otimes B)$ injective? i.e., How would we characterize such A and B? The most general possibility seems to be if it is true for all A and B that are not initial or final objects.

Answer 1

Your conditions don't seem to obtain very often, unfortunately.

Let's begin with the one-object case (where the one object is the unit, as it must be). This is the same thing as a monoid object in monoids, or equivalently, a commutative monoid $A$, by Eckmann-Hilton. Your requirement (1) appears to be that the multiplication $A\times A\to A$ is injective; requirement (2) is the same, applied to the invertible elements. This says that every element of $A$ can be uniquely factored, so that $A$ is trivial. So in the one-object case, (1) happens exactly once, and (2) is equivalent to the nonexistence of nontrivial invertible elements.

Using this, we deduce that the unit in a monoidal category satisfying either (1) or (2) is either very rigid in the sense that it admits no nontrivial endomorphisms or, respectively, rigid in the sense that it admits no nontrivial automorphisms.

If the category $\mathcal{C}$ has an initial object $\varnothing$ that is preserved by the monoidal structure (so that $X\otimes\varnothing=\varnothing\otimes X=\varnothing$ for all $X\in\mathcal{C}$), then every object has to (1) very rigid or (2) rigid. This kills off the bulk of the "algebraic" examples (like modules over a ring, etc.)

Also, note that your conditions are symmetric, i.e., (1) or (2) holds for $\mathcal{C}$ iff it holds for $\mathcal{C}^{\mathrm{op}}$. So Reid's example showing that not all cartesian categories satisfy (2) also works to show that not all cocartesian categories satisfy (2). (We contemplate the coproduct in $\mathrm{Set}^{\mathrm{op}}$...)

I don't want to be overly negative; so here's a situation in which I think your conditions do hold: suppose $\mathcal{C}$ a category with all finite coproducts. Suppose the morphism $X\to X\sqcup Y$ is monic for any $X,Y\in\mathcal{C}$. Then $\mathcal{C}$ (equipped with the coproduct) satisfies your condition. (So this works for finite sets with disjoint union, for instance.) Dually, if $\mathcal{C}$ is a category with all finite products such that the morphism $X\times Y\to X$ is epic for any $X,Y\in\mathcal{C}$, then $\mathcal{C}$ (equipped with the product) satisfies your condition. (So this works for the product on nonempty finite sets, for instance.)