nLab k-ary factorization system

-ary factorisation systems

kk-ary factorisation systems


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


For k>0k \gt 0 a natural number and CC a category (or \infty-category), a kk-ary factorisation system on CC consists of (k1)(k - 1) factorisation systems (E i,M i)(E_i, M_i) (for 0<i<k0 \lt i \lt k) on CC, such that

  • M iM i+1M_i \subseteq M_{i + 1} whenever this is meaningful (equivalently, E iE i1E_i \subseteq E_{i - 1}).

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

  • M 0M_0 consists of only isomorphisms/equivalences (equivalently, E 0E_0 consists of all morphisms), and
  • M kM_k consists of all morphisms (equivalently, E kE_k consists of only isomorphisms/equivalences).

Given a kk-ary factorisation system, the (co)image of (E i,M i)(E_i,M_i) is the ii-(co)image of the entire kk-ary factorisation system.

Note that every (higher) category has a unique 11-ary factorisation system, since no structure at all is required. We also say that a groupoid (or \infty-groupoid) has a (necessarily unique) 00-ary factorisation system; this makes sense since we have M 0=M kM_0 = M_k (and E 0=E kE_0 = E_k) in that case. A discrete category has a (necessarily unique) (1)(-1)-ary factorisation system.

A kk-ary factorisation system may also be called a kk-step factorisation system or a (k+1)(k+1)-stage factorisation system. You can see why if you count the basic morphisms (steps) and objects (stages) that k1k - 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) α\alpha. Then an α\alpha-stage factorisation system (in an ambient \infty-category CC) consists of an α\alpha-indexed family of factorisation systems (E i,M i)(E_i, M_i) in CC such that:

  • M iM jM_i \subseteq M_j whenever iji \leq j (equivalently, E iE jE_i \supseteq E_j whenever iji \leq j),
  • each morphism f:XYf\colon X \to Y is both the inverse limit limiim if\underset{i \to \infty}\lim \im_i f in the slice category C/YC/Y and the direct limit colimicoim if\underset{i \to -\infty}\colim \coim_i f in the coslice category X/CX/C, and
  • for each f:XYf\colon X \to Y, id Y\id_Y is colimiim if\underset{i \to -\infty}\colim \im_i f and id X\id_X is limicoim if\underset{i \to \infty}\lim \coim_i f.

This seems to be correct whenever α\alpha really is either an ordinal or the opposite thereof, as well as some other posets such as ω op+ω\omega^op + \omega (which is the poset of integers), but it seems to be missing something for (for example) ω+ω op\omega + \omega^op. Notice that, when α\alpha is both an ordinal and the opposite thereof, we recover the above definition of an (alpha1)(alpha-1)-ary factorisation system.



Last revised on January 23, 2021 at 18:09:22. See the history of this page for a list of all contributions to it.