# Monoidal category with initial object

Let $$(C, \otimes, 1)$$ be monoidal category with initial object $$0$$. Is it true that $$0 \otimes a \simeq 0$$ for all $$a \in C$$?

It is true in Set with cartesian product, $$R-$$modules with usual tensor product and $$\mathrm{End}(A)$$ the category of endofunctor of some category $$A$$ with composition. But is it true in general?

• It is not true. You can easily find some counterexamples for the dual question with terminal objects. Feb 8 at 15:03

Oh, I need to assume some compatibility of $$\otimes$$ with finite coproducts: $$\left( \coprod_{i} a_i \right) \otimes a \simeq \coprod_i (a_i \otimes a)$$. It is true in all those examples I gave. Now, $$0 \otimes a = 0$$ is just a tautology.
However, to answer your original question, a simple counterexample would be to take any cocartesian monoidal category; that is, take the tensor product to be given by the coproduct (e.g., abelian groups with direct sum, or sets with disjoint union). In particular, the tensor unit is given by the initial object in this case! This means that if $$\otimes$$ is given by the coproduct, then $$A\otimes0 = A$$ for every object $$A$$.