nLab k-ary factorization system

Idea

A $k$-ary factorization system is a generalization of (binary) orthogonal factorization systems and ternary factorization systems to factorizations into a string of $k$ morphisms.

Definition

For $k \gt 0$ a natural number and $C$ a category (or $\infty$-category), a $k$-ary factorisation system on $C$ consists of $(k - 1)$ factorisation systems $(E_i, M_i)$ (for $0 \lt i \lt k$) on $C$, such that

• $M_i \subseteq M_{i + 1}$ whenever this is meaningful (equivalently, $E_i \subseteq E_{i - 1}$).

We can extend this to include two other factorisation systems, one for $i = 0$ and one for $i = k$:

• $M_0$ consists of only isomorphisms/equivalences (equivalently, $E_0$ consists of all morphisms), and
• $M_k$ consists of all morphisms (equivalently, $E_k$ consists of only isomorphisms/equivalences).

Given a $k$-ary factorisation system, the (co)image of $(E_i,M_i)$ is the $i$-(co)image of the entire $k$-ary factorisation system.

Note that every (higher) category has a unique $1$-ary factorisation system, since no structure at all is required. We also say that a groupoid (or $\infty$-groupoid) has a (necessarily unique) $0$-ary factorisation system; this makes sense since we have $M_0 = M_k$ (and $E_0 = E_k$) in that case. A discrete category has a (necessarily unique) $(-1)$-ary factorisation system.

A $k$-ary factorisation system may also be called a $k$-step factorisation system or a $(k+1)$-stage factorisation system. You can see why if you count the basic morphisms (steps) and objects (stages) that $k - 1$ overlapping factorisation systems produce from a morphism.

Infinitary factorisation systems

Here is an incomplete attempt at a general definition:

Fix any ordinal number (or opposite thereof, or any poset, really) $\alpha$. Then an $\alpha$-stage factorisation system (in an ambient $\infty$-category $C$) consists of an $\alpha$-indexed family of factorisation systems $(E_i, M_i)$ in $C$ such that:

• $M_i \subseteq M_j$ whenever $i \leq j$ (equivalently, $E_i \supseteq E_j$ whenever $i \leq j$),
• each morphism $f\colon X \to Y$ is both the inverse limit $\underset{i \to \infty}\lim \im_i f$ in the slice category $C/Y$ and the direct limit $\underset{i \to -\infty}\colim \coim_i f$ in the coslice category $X/C$, and
• for each $f\colon X \to Y$, $\id_Y$ is $\underset{i \to -\infty}\colim \im_i f$ and $\id_X$ is $\underset{i \to \infty}\lim \coim_i f$.

This seems to be correct whenever $\alpha$ really is either an ordinal or the opposite thereof, as well as some other posets such as $\omega^op + \omega$ (which is the poset of integers), but it seems to be missing something for (for example) $\omega + \omega^op$. Notice that, when $\alpha$ is both an ordinal and the opposite thereof, we recover the above definition of an $(alpha-1)$-ary factorisation system.

Examples

• For $k = 3$ 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 $\infty$-ary factorization system consisting of the (n-epi, n-mono) factorization systems (the n-image-factorization) for all $n \in \mathbb{N}$. This is called the Postnikov system.

References

• Cafe discussion mainly on the ternary version

• Forum discussion including the k-ary case, even when k is infinite