We can extend this to include two other factorisation systems, one for and one for :
Note that every (higher) category has a unique -ary factorisation system, since no structure at all is required. We also say that a groupoid (or -groupoid) has a (necessarily unique) -ary factorisation system; this makes sense since we have (and ) in that case. A discrete category has a (necessarily unique) -ary factorisation system.
A -ary factorisation system may also be called a -step factorisation system or a -stage factorisation system. You can see why if you count the basic morphisms (steps) and objects (stages) that overlapping factorisation systems produce from a morphism.
Here is an incomplete attempt at a general definition:
Fix any ordinal number (or opposite thereof, or any poset, really) . Then an -stage factorisation system (in an ambient -category ) consists of an -indexed family of factorisation systems in such that:
This seems to be correct whenever really is either an ordinal or the opposite thereof, as well as some other posets such as (which is the poset of integers), but it seems to be missing something for (for example) . Notice that, when is both an ordinal and the opposite thereof, we recover the above definition of an -ary factorisation system.
For one speaks of a ternary factorization system. See there for more examples
In an (∞,1)-topos the (epi, mono) factorization system in a topos splits up to an -ary factorization system consisting of the (n-epi, n-mono) factorization systems (the n-image-factorization) for all . This is called the Postnikov system.
Cafe discussion mainy on the ternary version
Forum discussion including the k-ary case, even when k is infinite