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Started lift.
weak factorization system has redirects from: lifting property, right lifting property, left lifting property, lifting problem, lifting problems.
Would it be better to have these redirect to lift?
Thanks. True, the entire paragraph “Preliminaries” would deserve to have a dedicated entry. You could copy that over to “lift” and redirect the redirects accordingly.
Ok, have done that, but no doubt there’s a better arrangement at lift.
The lift of a morphism $f: Y\to B$ along an epimorphism (or more general map) $p:X\to B$ is a morphism $\tilde{f}: Y\to X$ such that $f = p\circ\tilde{f}$.
I changed “the” here to “a”.
Looking at extension where it has dually
The extension of a morphism…
it doesn’t seem so clearly wrong. I guess there’s ambiguity between ’extension’ as the act of extending and as an instance of so extending, as you see in the second paragraph of lift
The dual problem is the extension of a morphism…
I’ll change ’the’ to ’a’ at extension.
Following discussion in another thread here, I have created a new section
and filled it with material that, up to some superficial first polishing by me, has kindly been written by the Anonymous Guest from that thread (copied over from Sandbox rev 2402, where it was first compiled as Sandbox rev 2401).
As I write in that other thread, I think there is much room to polish this material up and make it more readable and more use-friendly, but I think it’s great to have this considerable list of examples now. Maybe it inspires more readers here to join in and try their hand at some editing.
At present the Idea section starts
Intuitively, the lifting property is a negation in category theory:
This seems strange to me. That does not fit my intuition at all! No doubt there are examples of lifting properties that are somehow a bit like some form of negation, and a lot of others a bit like ’orthogonality or ’being perpendicular to’ but it does seem a bit strange to start like this. There are also important examples that do not correspond intuitively to a negation at all.
Excellent. That looks much clearer and I now understand where the ‘negation’ idea was coming from.
I found myself musing for a moment how general this ’negation’ point of view is, for instance whether it holds for the standard model structures on simplicial and cubical sets…I think it does! One can I think reasonably argue that horn inclusions are the correct analogues of $\emptyset \rightarrow *$ in higher dimensions, so it works out correctly that Kan fibrations are the ’negations of archetypical cofibrations’, and I think it is also true that cofibrations are the ’negations of archetypical Kan fibrations’, though I’ve never seen this expressed before: I think the obvious analogues of the map from a two point set to a one point set in higher dimensions would be the fold maps $\square^{n} \sqcup \square^{n} \rightarrow \square^{n}$ for all $n$ (for cubical sets; the obvious analogue for simplicial sets), and, as far as I see, a) these are indeed Kan fibrations b) cofibrations (i.e. monomorphisms) are exactly those morphisms of cubical/simplicial sets with the right lifting property with respect to the set of these fold maps ranging over $n \geq 0$. I wonder if this observation is good for anything!
Yes, I gather such observations are the reason for Misha Gavrilovich’s choice of terminology “Quillen negation” (though that entry remains a stub and the respective comments in the article listed there are brief).
The whole development of separation axioms in terms of lifting properties is driven, I feel, by a sense of wonder that some basic but neat observations about lifting properties have remained underappreciated.
This reminds me a lot of Categories, Allegories where Freyd and Scedrov formulate a version of first-order logic of diagrams that was centered around looking at how a diagram $J_0 \to C$ extends along a functor $J_0 \to J_1$. For example, $1 \to C$ is an initial object iff every extension along $1 \to 1+1$ (inserting into the left term) has a unique extension along $1+1 \to [1]$ (inserting as source and target).
(I’m more reacting to “formulate things as lifting properties” in general, rather than an idea of ’negation’)
Perhaps Misha will see this thread and care to comment. If there were some way to construct a model structure using these kind of observations, or use the characterisation of the cofibrations that I gave in #11 for some interesting purpose, that would be very nice.
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