In a pointed category $C$, for any family $\{C_i\}_{i \in I}$ of objects, such that both their product and their coproduct exist, there is a canonical morphism
$$
\varphi: \coprod\limits_{i \in I} C_i \to \prod\limits_{i \in I} C_i
$$
defined by the components
$$
\text{pr}_j \circ \varphi \circ \text{ins}_i = \delta_{ji}
$$
for all $i,j \in I$, where $\delta_{ji}$ is the identity if $i = j$ and the zero morphism otherwise. If this morphism is an isomorphism for finite $I$, the (co) product is called a biproduct, but apart from this important case I find very little information about it in the literature.
By many examples, I see the following for finite $I$:
- $\varphi$ can be monic (e.g. $\text{Set}_*, \text{Mod}_R$) or not ($\text{Group}$).
- $\varphi$ can be epic (algebraic categories like $\text{Group}, \text{Mod}_R$ etc.) or not (e.g. $\text{Set}_*$).
- $\varphi$ can even be an isomorphism (e.g. $\text{Mod}_R$).
Among the possible combinations, there are two that I could not find. These constitute my questions:
- Is there a case where $\varphi$ is neither epic nor monic?
- Is there a case where $\varphi$ is a non-isic bimorphism?
Best Answer
Yes, there's sort of a lame way to do this by taking the product of two of your existing examples, e.g. $\text{Group} \times \text{Set}_{\ast}$.
Yes. Consider the category $\text{Ban}_1$ of Banach spaces and weak contractions (morphisms of norm $\le 1$). The finite coproduct and product in this category are both the direct sum, but with different norms: the finite coproduct has an "$\ell^1$ norm" and the finite product has an "$\ell^{\infty}$ norm." The underlying map of sets from the coproduct to the product is a bijection, so this map is a bimorphism, but it's not an isomorphism.