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Created factorization system in a 2-category.
Is there any call for a factorization into a triple of systems, as in the stuff, structure, property story?
I don’t think I’ve ever seen that situation formalized, but it would be worth doing. I can think of at least one other example.
The other example being the combination of (hyperconnected, localic) and (surjection, inclusion) factorizations for topoi, which are very similar to the (eso+full, faithful) and (eso, fully faithful) factorizations for categories.
In a recent email exchange with Andre Joyal he pointed out to me the factorisation systems
(epimorphisms of toposes, sub-toposes)
(connected morphisms of toposes, totally disconnected localic morphisms)
(1-connected morphisms of toposes, Galois morphisms)
Are these examples of factorisation systems in a 2-category (I mean interestingly so)? I believe the third system has only be looked at in the special case of morphisms $E \to Set$. Marta Bunge I think has done the second, and the first one is ’old’.
Mike @3: Does any notion of orthogonality exist for triple factorisations? Would that matter to their potential interest? Maybe what I’d really like to know is the whole point of factorisation systems.
DavidR #5, the first one is definitely a factorization system, aka (surjections, inclusions). I’m not sure about the others, but it seems quite possible. I know there’s an “almost” factorization system on toposes which consists of (connected+locally-connected morphisms, local homeomorphisms) but according to the Elephant the factorizations only exist for morphisms that are already known to be locally connected. Perhaps the second one you mention above is a version of this generalized to non-locally-connected morphisms.
@DavidC #6: Both examples of triple factorizations are “actually” just two pairs of ordinary “double” factorizations related in a certain way, so they have two resulting kinds of orthogonality.
The Whole Point of Factorization Systems would probably be a good topic for a blog post one day… (-: I’ll see if I can think of some good examples.
The Whole Point of Factorization Systems would probably be a good topic for a blog post one day
Yes. Or also for an Idea-section in the relevant nLab entry. It is one of these concepts whose impact is in maybe surprising contrast to the simplicity of their definition. I am not sure if I understand The Whole Point of factorization systems. I think I understand some points of them.
@2: David, when I hear the word 2-categorical factorization system my first association is to split it into 3 as in the Postnikov tower for 2-types. As I was discussing before this could play a role in a possible discussion of model 2-categories which would model $(\infty,2)$-categories. I am very interested in this idea which may be a blunder despite its intuitive plausibility.
@7: Urs, in joyalslab, there is a beautiful, original and extensive treatment of factorization systems and weak factorization systems.
in joyalslab, there is a beautiful, original and extensive treatment of factorization systems and weak factorization systems.
That reminds me: we need to start putting links to Joyal’s CatLab into the relevant nLab enties.
@9: Zoran, so instead of the 3 classes for a model category forming weak factorisation systems $(C, F \cap W)$ and $(C \cap W, F)$ , there might be 5 classes with various combinations of intersections making weak factorisation triple systems? Or something like that.
@David #11
or 4 classes? Weak equivalences $W$ + three other classes $(A,B,C)$, the intersection of $W$ with each of the others in turn to form $(A\cap W,B,C)$ etc. to get the three classes needed to form a three-way factorisation?
added to factorization system in a 2-category a reference (there are probably more canonical ones, though) and a remark that the system [essentially srujective + full] / [faithful] on Grpd is a special case of the n-connected/n-truncated factorization system. I’ll now add more details to that latter entry.
Wouldn’t this article be better called “factorization system for a 2-category” or “factorization system on a 2-category”? Usually “in” is used to describe internalisations of Cat-concepts in some other 2-category (e.g. monad in a 2-category). “on” refers to the 2-dimensional generalisation (e.g. 2-monad on a 2-category). I would expect “factorization system in a 2-category” to refer to a notion of factorisation for objects in the 2-category, not 1-cells of the 2-category itself.
[Administrative note: merged #14, which was in a new thread, into an old thread, and re-named the old thread so that it is correctly detected as the discussion thread for the nLab page of the same name.]
“factorization system on a 2-category” is already a redirect, and “for” could certainly also be. I don’t really care what the page itself is named; I don’t think there’s much danger of confusion since for the internal notion I would tend to say instead “factorization system on an object of a 2-category”.
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