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.

]]>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”.

]]>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.

]]>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.

]]>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.

]]>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)?

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