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Created essentially pointwise isomorphic, with a reference.
Is this terminology anywhere in the literature, or did you make it up? It reads as if you made it up.
Not used before afaik. Invented in an attempt to systematize.
I am curious where you are heading with these little articles? I’m not seeing any real purpose to essentially pointwise isomorphic, certainly not in its current state.
(By the way, since you seem to be planning on further activity on the nLab, may I recommend that you spend a little time looking at HowTo? For example, our naming conventions ask that page titles be singular nouns – this actually should be amended now to “noun phrases” or something since we have recently created a lot of pages titled by brief theorem-statements. So the currently named pages with adjective titles will probably be retitled. Also you will find some useful information on linking and cross-referencing; some of your recent additions need to be fixed up a little in this regard.)
Getting back to pages like unnaturally isomorphic: as I say I’m not sure where you’re headed. As a matter of pedagogy it’s not a bad idea to give non-examples or counterexamples along with examples; it’s what should be done in the classroom for instance, and one should be aware of what can go wrong. If that’s all you want these pages for, then that’d be ok. But as a concept in its own right, I don’t see that “unnaturally isomorphic” plays much of a role in developing category theory; indeed, we tend to carefully avoid such circumstances and focus on “what goes right” in category theory. So writing a bunch of pages on it seems like maybe sending an odd message, hence my question.
Thanks. Had a look at the HowTo at the very beginning, but not for long. Will take a second look.
The intention of unnaturally isomorphic is to improve appreciation and understanding of natural isomorphism and similar articles. Not as a concept to be explored in its own right. But since you are concerned about it I will stop writing it for the time being.
Well, I don’t mean to give you a hard time about it, and sorry if it came across like that. It’s just that I never hear anyone say “two functors are unnaturally isomorphic” (either as hypothesis or as conclusion), although I do hear “these two functors are not naturally isomorphic”, which has a rather different ring to it. For example, the species of bi-pointed trees and the species of permutations are not naturally isomorphic. Category theorists may even drop the “naturally” and just say the functors are isomorphic, without any confusion, because isomorphism in a functor category means (by definition of functor category) a natural isomorphism – morally there is no other kind.
Never thought of bi-pointed trees, but I know species a bit. Your example reminds me of another (or is it?) example: there does not exist a natural isomorphism between the species of linear orderings and the species of permutations.
Oh sorry, I said that wrong. I should have said the species of bipointed trees not being isomorphic to the species of endofunctions, my bad. And yes, those two sets of examples are intimately related! The species of bipointed trees is isomorphic to the species composition , whereas the species of endofunctions is isomorphic to the species composition .
I think a discussion of unnatural isomorphisms is not out of place. There are interesting examples, such as the functors back and forth between vector spaces and affine spaces, whose composite on the vector-space side is naturally isomorphic to the identity, but whose composite on the affine-space side is unnaturally isomorphic to the identity, and therefore the two categories are not equivalent. It’s not something one would tend to prove theorems about, but I think looking at some examples of unnatural isomorphisms can indeed improve appreciation for natural ones.
But I don’t see any purpose to inventing a new “possible synonym” for a term that is rarely used anyway, so I would suggest deleting “essentially pointwise isomorphic”.
@Mike Shulman: I agree, essentially pointwise isomorphic was more or less an accident in an attempt to systematize, and would also suggest deleting this article.
Please, do not delete unnatural isomorphism for the time being though. I have a clearer conception of a concise expository discussion of this topic now, in particular, sharpening the article by focusing on
unnaturally isomorphic functors
only, and removing the explicit itemized distinction between
unnaturally isomorphic
objects
categories
functors
again.
In particular, because
unnaturally isomorphic objects
are a special instance of
unnaturally isomorphic functor
because (roughly speaking, not enough time at the moment)
objects O and O’ of a category C are unnaturally isomorphic if and only if
there exists an endofunctor F of C which
is unnaturally isomorphic to the identity-endofunctor of C
sends O to O’
This in particular formalizes the usual example of V being unnaturally isomorphic to V*, by taking F to be the dualize-once-endofunctor, which satisfies the two conditions above.
Again, I intend to improve and sharpen the article, which indeed can serve to better appreciate natural isomorphism’, but the world kept intervening, and this will have to wait till tomorrow.
Incidentally, inside the article I will briefly suggest a synonym
statically isomorphic
which seems mnemonically helpful for learners, but won’t create an article for this term.
I don’t think it makes any sense to speak of two objects being unnaturally isomorphic. Just like “naturality” is only sensical for isomorphisms between functors, likewise so is “unnaturality.”
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