David Roberts (n+1)-ary factorisation system on n Cat

The category Set has a binary factorisation system (epi,mono)The category Cat has a 3-ary factorisation system (full+eso,faithful+eso,full+faithful)

Generally, there should be an $(n+1)$-ary factorisation system on $nCat$.

See for example this comment and discussion following. The proposal in comment 10 (reproduced below) should be shown to satisfy (at least one of) the definitions from comments 25, 28.

Comment 10:

I have an idea how to do this. Take for a test case 1-categories. For a category $C$ map $X \to Ob(C)$ let $C[X]$ be the category given by the pullback $codisc(X)\times_{codisc(Obj(C))} C$ (In other words, objects $X$ and arrows $X^2\times_{Obj(C)^2} Mor(C)$). Then for a functor $f:D \to C$ there is a factorisation

$$D \to C[Obj(D)] \to C[im(f_0)] \to C$$

where $f_0$ is the object-component of $f$.

A similar game can be played with 2-categories, where one can not only define $C[X]$ for a set $X$, but also a reflexive graph $X_\bullet = (X_1 \rightrightarrows X_0)$ with composition (something like a (pointed magma)-oid :) and a map of such things $X_\bullet \to C_{\le 1}$ for the underlying 1-category $C_{\le 1}$ of a 2-category (this works for bicategories too, but let’s stick to strict things). In particular, for the underlying category $D_{\le 1}$ of a second 2-category $D$ equipped with a 2-functor $F:D \to C$. In general this is a bicategory, but if $X_\bullet$ is a category, then $C[X_\bullet]$ is a 2-category.

We can then define

$$D \to C[D_{\le 1}] \to C[D_{\le 0}] \to C[im(F_0)] \to C$$

Each of the things $C[?]$ expresses that $2Cat$ is fibred (opfibred?) over various other categories and they can all be defined in terms of pullbacks with variously codiscrete 2-categories(1). This should make it manifest what sort of things are forgotten (stuff, structure etc) at each step. The universal property of the pullbacks helps ensure the uniqueness up to isomorphism of the factorisation.

The general pattern should be clear. There is a functor $codisc_{n+1}: nCat \to (n+1)Cat$ adding to each category a unique arrow between any two parallel n-arrows. For an $n$-functor $D \to C$ define $C[D_{\le m}] = codisc_{m+1}(D_{\le m}) \times_{codisc(C_{\le m})} C$ where the $(m+1)$-codiscrete $(m+1)$-categories are considered $n$-categories with only identity arrows between dimensions $m+1$ and $n$. Here $m=0,\ldots,n-1$. The successive truncations D,D_{le (n-1)),D_{le (n-2)),\ldots,D_{le 0) together with the universal property of the pullback should furnish the functors in the factorisation.

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(1) There is a functor $codisc_2:rGrp_{comp} \to Bicat$ from reflexive graphs with composition to $Bicat$ by adding to a reflexive graph a unique 2-arrow between any two parallel 1-arrows. Coherence happens automatically. The restriction of $codisc_2$ to $Cat \hookrightarrow rGrp_{comp}$ lands in $2Cat \hookrightarrow Bicat$.

Comment 25 (Toby Bartels):

For $n \gt 1$, I claim that an $n$-ary factorisation system consists of $n^+$ (that is $n + 1$) factorisation systems $(E_i,M_i)$ (for $0 \leq i \leq n$) such that

• $M_i \subseteq M_{i^+}$ for $0 \leq i \lt n$ (equivalently, $E_i \supseteq E_{i^+}$ for $0 \leq i \lt n$),
• $M_0$ consists of only isomorphisms/equivalences (equivalently, $E_0$ consists of all morphisms), and
• $M_n$ consists of all morphisms (equivalently, $E_n$ consists of only isomorphisms/equivalences).

(Or course, an $n$-ary factorisation system is determined by the $n - 1$ factorisations systems $(E_i,M_i)$ for $0 \lt i \lt n$, but the the other two exist.) Do you agree?

Given an $n$-ary factorisation system, the (co)image of $(E_i,M_i)$ is the $i$-(co)image of the entire $n$-ary factorisation system. (This agrees with the terminology in $Cat$ for $n = 3$, or more generally with the terminology in $(n - 2) Cat$ or even $(\infty,n - 2) Cat$.)

Then extending this definition to lower values of $n$, every category (or $\infty$-category) has a unique $1$-ary factorisation system, where $(E_0,M_0)$ is (iso,all) and $(E_1,M_1)$ is (all,iso), as you suggested.

A category has a $0$-ary factorisation system if and only if it is a groupoid, in which case $(E_0,M_0)$ is both (iso,all) and (all,iso) at once. In other words, rather than requiring every morphism to be a $0$-ary composite on the nose, we require every morphism to be a $0$-ary composite up to isomorphism. I think that this is right, since a factorisation system (of any arity) should be given by specifying full and replete subcategories of the arrow category (or equivalently, collections of isomorphism classes of the arrow category), and every isomorphism is isomorphic to an identity.

Comment 28 (also Toby Bartels):

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 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 $\underset{i \to \infty}\lim \coim_i f$.