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I corrected a couple og microscopic typos at k-ary factorization system, and then I noticed that something is unclear in the definition: first of all the family of factorization system is asked to be strong (= uniqueness of solution to any lifting problem) or weak (existence, no uniqueness)? And when the definition says
$M_1 \subseteq \dots \subseteq M_{\kappa-1}$ whenever this is meaningful (equivalently, $E_{k-1} \subseteq\dots\subseteq E_{1}$)
what does it precisely mean? Are we asking that right classes be nested?
Thirdly, it is my humble opinion that saying
A discrete category has a (necessarily unique) $(-1)$-ary factorisation system.
is formally incorrect: discrete categories are groupoids where the only arrows are identities, so this is a particular kind of 0-ary factorization system.
Instead, negative thinking suggests that (-1)-ary factorization systems live in non-unital categories, and detect precisely the case where the class of isomorphisms is empty (recall that in a WFS $(L,R)$ the intersection $L\cap R$ consists of all isomorphisms; if in a 0-ary factorization system we had $L=R=L\cap R=Iso(\mathbf C)$, morally in a (-1)-ary system the intersection has to be empty, giving a category without identities -i.e. a particular kind of “plot”, in the jargon of this paper which I finally convinced my friend Salvatore to put on the arXiv-, and more precisely an associative, “strongly nonunital” plot).
This leads to another question: how can be the notion of (W)FS be extended to Mitchell’s semicategories (with empty or partially defined identity function)?
Of course, we can consider both orthogonal and weak k-ary factorization systems. Perhaps it would be clearest to write the main entry about orthogonal ones, and then mention later that there is a weak analogue?
I think “whenever this is meaningful” means “whenever $i\gt 0$ and $i+1\lt k$” so that $M_i$ and $M_{i+1}$ are defined.
Toby was the one who wrote the bit about the (-1)-ary case, so maybe he can respond to that. Personally I can’t yet see how to make any sense of the (-1)-ary case.
In fact I perfectly [well, more or less…] see how one can define (-1)-ary case, but we have to move to the far more general setting of nonunital (or even nonassociative, partially defined) categories. I think that in some sense this is studied in Salvatore’s work (see in particular Example 9 and Remark 16), since after a short discussion he seconds my sensation.
I’m not convinced by the mere argument “in the other cases, the interersection is the isomorphisms, and thus in this case the isomorphisms have to be empty”. The general definition of k-ary factorization system lives on a category; I would need more motivation to be convinced that in the particular case $k=-1$ we have to change the type of the definition.
Also, I retract my comment in #2 that we can consider weak k-ary factorization systems. We can consider sequences of nested weak factorization systems, but I don’t think I would call them k-ary factorization systems, since they don’t give rise to unique k-ary factorizations of morphisms.
Hmm, maybe I have an idea of what Toby had in mind, but it leads me to think that it’s 0-ary factorization systems that should live on discrete categories. Namely, the point of a k-ary factorization system is that every morphism factors uniquely as a composite of k factors, and a composite of 0 factors is just an identity morphism — while I can’t see how to make any sense of “a composite of (-1) factors”.
Just noting here as I stumbled across this, that discussion of creation of this page and some early updates to it can be found at the discussion thread for ternary factorization system.
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