Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I was sure there was a discussion to remove this page, because it was agreed that being balanced was not a property that was relevant in category theory. However, I can’t find the discussion now… I would have expected it to be in this thread, so I’m surprised there weren’t any previous messages here.
How is it not relevant?
added a couple of references:
Peter Johnstone, Cor. 1.22 in: Topos theory, London Math. Soc. Monographs 10, Acad. Press (1977), Dover (2014)
Roy L. Crole, p. 115 of: Categories for types, Cambridge University Press (1994) [doi:10.1017/CBO9781139172707]
Regarding relevance, it’s quite telling that there are many examples on this page, but no properties or results involving balanced categories. That may just be because the page has not been developed significantly, but my feeling is that it is primarily because there are very few interesting properties or results regarding balanced categories. This answer by Mike Shulman implies that he knows of no uses of the property, and Todd Trimble suggests a single one: the result that a balanced quasitopos is a topos. For a concept to be worth a name, one should expect at least a few useful results. It would be nice to find some others.
It sounds like you are overthinking this.
We clearly want to be able to write things like:
To prove that is an isomorphism it is now sufficient, since is balanced, to prove that it is both a monomorphism and an epimorphism.
Also, the discussion you point to argues against saying “bimorphism” not against “balanced category”.
I don’t think this entry should be deleted.
We clearly want to be able to write things like:
This kind of argument arises commonly in the consideration of factorisation systems on categories: one often sees a proof appealing to the fact that a morphism that is both in the left class and the right class of a factorisation system is an isomorphism. However, making use of such an argument in a category without such a factorisation system, in the specific case that E = { epis } and M = { monos }, seems uncommon (I only know of the one aforementioned example). However, it could be that I am not familiar with literature that makes use of these kinds of arguments. That is why I think it would be nice to write down some examples on this page, if anyone knows of any.
Also, the discussion you point to argues against saying “bimorphism” not against “balanced category”.
The very first paragraph of the discussion argues against the value of “bimorphism”, but the rest of the discussion is a counterargument to the claim that “the concept of bimorphism is valuable because the concept of balanced category is valuable”.
It sounds like you are overthinking this.
Perhaps. It has just always struck me as odd that many authors define the notion of balanced category (e.g. in introductory textbooks), and give examples and counterexamples of such, yet rarely, if ever, make use of the concept technically.
(Re.:
I was sure there was a discussion to remove this page, because it was agreed that being balanced was not a property that was relevant in category theory. However, I can’t find the discussion now… I would have expected it to be in this thread, so I’m surprised there weren’t any previous messages here.
I realised I had been thinking of the nLab page on bimorphisms (https://nforum.ncatlab.org/discussion/13659/bimorphism), which was removed.)
the page currently has the example:
In a poset or a quiver, or more generally in any thin category, every morphism is both monic and epic
which doesn’t make sense since a quiver isn’t a category. Maybe “free category on a quiver” was intended but such is not generally thin.
I have reworded the last (counter-)example to clarify the intended meaning (as far as I can tell, this is from revision 9).
(Incidentally, in the page history, starting with that revision 9, one can see the old discussion, inside a query box, that the comment #2 above may have been remembering.)
1 to 11 of 11