Canonical morphism from coproduct to product – Questions 3-6: Infinite index sets

category-theory

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 infinite $I$:

  • $\varphi$ can be monic (e.g. $\text{Set}_*, \text{Mod}_R$) or not ($\text{Group}$).
  • $\varphi$ is commonly not epic (the only counterexample I know of is the (co)product of zero objects).

These these yield my questions:

  1. Is there a case where $\varphi$ is epic with nontrivial (co)factors? If there is:

  2. Are there cases where (for nontrivial cofactors) $\varphi$ is epic and

    • not monic.
    • monic but not isic.
    • even isic.

Can we make some triviality assertions on $C$, if one of these hold?

Best Answer

  1. Yes, take the opposite of a pointed category where you know it's monic, e.g. $\text{Ab}^{op}$. By Pontryagin duality this is equivalent to the category of compact Hausdorff abelian groups. The product is the cartesian product but the coproduct is compactification of the direct sum.

  2. As above, in $\text{Ab}^{op}$ it's epic but not monic. I'm going to separate out your other two subquestions as full questions.

  3. For an example which is both monic and epic but not an iso I think we can again consider the category $\text{Ban}_1$ of Banach spaces and weak contractions. It's worth spelling out in some detail what the product and coproduct are here. For a family $X_i$ of Banach spaces, the coproduct is the completion of the direct sum under the "$\ell^1$ norm" $\| (x_i) \|_1 = \sum_i \| x_i \|$. The product is the subspace of the cartesian product for which the "$\ell^{\infty}$ norm" $\| (x_i) \|_{\infty} = \sup_i \| x_i \|$ is finite. I think the map $\coprod_i X_i \to \prod_i X_i$ is injective (but I haven't checked this) with dense image (note that the norm on the codomain is not the induced norm on the image), which means it's both monic and epic.

  4. For an example which is even an iso I think we can consider the category of suplattices. I believe, but have not checked, that coproducts and products of arbitrary arity agree here. This is at least consistent with the nLab's claim that the free suplattice on a set $X$ (equivalently, the coproduct of $|X|$ copies of the free suplattice on a singleton) is the powerset $2^X$, which is also the product of $|X|$ copies of the free suplattice on a singleton.

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