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I looked at a number of entries like zero morphism, direct sum, biproduct, kernel etc. I am writing some lecture on additive categories.
Here is some confusion t seems. $Ab$-categories are categories enriched over the category of Abelian groups with monoidal product being the tensor product. These entries talk about how to define zero morphism, kernel (as representing object) etc. for such as a special case of enrichement over pointed sets ! But it does not look right to me. Pointed sets are assumed in zero morphism to be monoidal with respect to the smash product of sets. But the tensor product of Abelian groups after forgetting the Abelian structure to pointed sets, unless I am halucinating, disagrees with the smash product of the underlying pointed sets! So the general treatment which would be good enough for enrichement over pointed sets, Abelian groups and few others should be done more carefully, it seems. Any thoughts ?
It is also unclear how standard is our definition at additive and abelian categories that preadditive categores are the $Ab$-categories with zero object. MacLane in Homology (1963) lists the axioms for additive categories and then remarks that if one does not assume direct sums (but he defines just binary direct sums, so it does not include the existence of zero object) that this is called preadditive. In this reading it means we agree with MacLane. Weibel does not use the term preadditive in his book. Popescu (and wikipedia which follows it) as well as Borceux call preadditive the same as $Ab$-enriched.
Also archiving a query box from biproduct (with a backpointer):
Mike: Can anyone give a definition of a biproduct that doesn’t require the category to be presupposed to have a zero object, but which specializes to a zero object in the 0-ary case?
Toby Bartels: Actually, our current definition does this; I just wrote it badly. Of course, now the category starts with an enriched structure … but at least it is a weaker requirement.
Mike: The reason I ask is that I’m trying to work out an indexed version of biproducts. In a fibration or indexed category, coproducts are left adjoint $f_!$ to reindexing $f^*$ and products are right adjoint $f_*$, so having $f$-indexed biproducts should mean that some canonical map $f_!\to f_*$ is an isomorphism. But I’m not having much luck constructing such a canonical map. Perhaps this is related to the need for decidable equality?
Toby: If it helps, here is a bit more on that subject, which I wrote at direct sum:
An arbitrary index set will still work if $C$ is enriched over the category of sets and partial functions; this may be embedded as a full subcategory of the category of pointed sets, and the embedding is an equivalence of categories if and only if the law of excluded middle holds. But the usual examples of $C$ are not (constructively) so enriched.
Mike: Was that what you had in mind for the “extra structure” above? It occurred to me that it could also mean the existence of “subzero objects,” i.e. $\coprod_A X \to \prod_A X$ is an isomorphism whenever $A$ is a subsingleton—which is also, I think, not constructively true in the usual examples.
Toby: Yes, that's what I had in mind. (Although you could also do a more local version, letting the requirement that the index set be discrete and the requirement the hom-sets be enriched over $Set_part$ meet halfway, if you see what I mean.)
At related entry generalized kernel, there is some ill parsed code which has ulconer urcorner etc. few times in the middle section. Also a spurious link to invertee (should be inverter ?).
Here is some confusion t seems. $Ab$-categories are categories enriched over the category of Abelian groups with monoidal product being the tensor product. These entries talk about how to define zero morphism, kernel (as representing object) etc. for such as a special case of enrichement over pointed sets ! But it does not look right to me. Pointed sets are assumed in zero morphism to be monoidal with respect to the smash product of sets. But the tensor product of Abelian groups after forgetting the Abelian structure to pointed sets, unless I am halucinating, disagrees with the smash product of the underlying pointed sets! So the general treatment which would be good enough for enrichement over pointed sets, Abelian groups and few others should be done more carefully, it seems. Any thoughts ?
The left adjoint to the forgetful functor is strong monoidal, so there’s a canonical map $A\wedge B\to A\otimes B$ which makes any $Ab$-enriched category enriched in pointed sets.
I’d add to Marc’s comment that the relevant adjunction is an adjunction in the 2-category of symmetric monoidal categories, (lax) symmetric monoidal functors, and monoidal transformations. In that circumstance, the left adjoint is necessarily strong symmetric monoidal. This is extremely typical for change-of-base between enriched categories.
I don’t think there is any confusion. The forgetful functor, being the right adjoint in this situation, preserves weighted limits (such as kernels). (It also reflects isomorphisms, so it also reflects weighted limits.) And so weighted limits in abelian groups may be calculated as in pointed sets. (Weighted colimits are of course another story, in general.)
Terminology about ‘preadditive’, ‘additive’, etc is not consistent in the literature. Even ‘abelian’ took a while to nail down. I think that we use a good system, one that gives a term to everything that is wanted and uses the simpler terms for what is most wanted. But some references will disagree.
It might be a good idea for the pages to mention this monoidal adjunction, though, for the benefit of anyone else feeling the same confusion.
Thanks, guys, nice to have the experts around :) About additive, I think nowdays at least additive (and abelian) are standard. I thought to write an entry on historical and terminological aspects accompanying additive and abelian categories.
Re #7: this could take the form of an nLab article on “change of base”, perhaps? We do have an article base change, but this seems to be mostly about change of base functors between slice categories, and the context could and should be widened to encompass change of base between enriched categories, etc., perhaps following the approach taken in Dominic Verity’s thesis (see here.
I very much agree with Todd #9.
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