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I have been polishing and expanding the entry weak factorization system:
gave it an Idea-section;
gave the definitions numbered environments and full details;
spelled out the proof of the closure properties in full detail.
Regarding notation: I decided to use as generic name for a weak factorization system:
not $(L,R)$, as used to be used in the entry (for that’s already my preferred generic choice for pairs of adjoint functors on the $n$Lab, and for discussion of Quillen adjunctions the notation conventions would clash);
not $(E,M)$ or the like, since that gives no hint as to what is meant (running into an “$E$” in the middle of some discussion, the reader is always at risk of having to browse back to figure out which class is meant);
but… $(Proj, Inj)$, for that is nicely indicating what is meant.
Does anyone else use that terminology/notation?
Given a single class $K$ of morphisms, it’s standard to write $K Proj$ and $K Inj$ for the corresponding left and right lifting classes. Accordingly, it’s standard to say things like “$(K,K Inj)$ happens to be a weak factorization system” or $(K Proj,K )$ happens to be a weak factorization system”.
I much prefer $(L,R)$. I generally use $F\dashv G$ for a generic pair of adjoint functors, but maybe you could do $(\mathcal{L},\mathcal{R})$ to disambiguate?
Let’s maybe wait with changing notation back until edits stabilize.
Because, for instance that statement about the closure property is all about the lifting axioms and independent of the factorization, so its proof might better be copied over to the entry on lifting problems. There notation such as $(\mathcal{L}.\mathcal{R})$ wouldn’t work, while $Inj$ and $Proj$ would be preferred there.
In my opinion, notation such as $Inj$ and $Proj$ would never be preferred.
Why?
For one thing, I think it’s general mathematical practice that variables have single-character names. Multi-character identifiers are becoming more common than they used to be, but in all cases I can think of they are used to denote specific, defined objects rather than variables being quantified over. We talk about “the category $Set$” but we say “for any category $C$”. I think violating a convention like this would be very confusing.
My experience is the opposite, frequently when looking up things in a longer text I need to waste time with browsing to figure out what the authors means by their letters. Notation that comes with indication of its meaning is less confusing and more efficient for communication. And specifically things like “fib, cof, proj, inj” are standard.
But let’s leave it at that. I have replaced the $(Proj,Inj)$ in the entry now with $(\mathcal{L},\mathcal{R})$.
fib, cof, proj, inj are standard as operations that are specific and defined, and that are applied to variables, e.g. $cof(I)$ or $inj(J)$.
not sure why we really need to debate this. I find myself now interrupting doing useful work and instead collecting references just to argue about notation; I shouldn’t do that. But before I could stop myself, the first hit was this here: pdf.
But let’s really leave it at that. I have changed the notation on the nLab to your preference, so all is good.
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