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)(n+1)-ary factorisation system on nCatnCat.

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 CC map XOb(C)X \to Ob(C) let C[X]C[X] be the category given by the pullback codisc(X)× codisc(Obj(C))Ccodisc(X)\times_{codisc(Obj(C))} C (In other words, objects XX and arrows X 2× Obj(C) 2Mor(C)X^2\times_{Obj(C)^2} Mor(C)). Then for a functor f:DCf:D \to C there is a factorisation

DC[Obj(D)]C[im(f 0)]C D \to C[Obj(D)] \to C[im(f_0)] \to C

where f 0f_0 is the object-component of ff.

A similar game can be played with 2-categories, where one can not only define C[X]C[X] for a set XX, but also a reflexive graph X =(X 1X 0)X_\bullet = (X_1 \rightrightarrows X_0) with composition (something like a (pointed magma)-oid :) and a map of such things X C 1X_\bullet \to C_{\le 1} for the underlying 1-category C 1 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 1D_{\le 1} of a second 2-category DD equipped with a 2-functor F:DCF:D \to C. In general this is a bicategory, but if X X_\bullet is a category, then C[X ]C[X_\bullet] is a 2-category.

We can then define

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

Each of the things C[?]C[?] expresses that 2Cat2Cat 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(n+1)Catcodisc_{n+1}: nCat \to (n+1)Cat adding to each category a unique arrow between any two parallel n-arrows. For an nn-functor DCD \to C define C[D m]=codisc m+1(D m)× codisc(C m)CC[D_{\le m}] = codisc_{m+1}(D_{\le m}) \times_{codisc(C_{\le m})} C where the (m+1)(m+1)-codiscrete (m+1)(m+1)-categories are considered nn-categories with only identity arrows between dimensions m+1m+1 and nn. Here m=0,,n1m=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.

(1) There is a functor codisc 2:rGrp compBicatcodisc_2:rGrp_{comp} \to Bicat from reflexive graphs with composition to BicatBicat by adding to a reflexive graph a unique 2-arrow between any two parallel 1-arrows. Coherence happens automatically. The restriction of codisc 2codisc_2 to CatrGrp compCat \hookrightarrow rGrp_{comp} lands in 2CatBicat2Cat \hookrightarrow Bicat.

Comment 25 (Toby Bartels):

For n>1n \gt 1, I claim that an nn-ary factorisation system consists of n +n^+ (that is n+1n + 1) factorisation systems (E i,M i)(E_i,M_i) (for 0in0 \leq i \leq n) such that

  • M iM i +M_i \subseteq M_{i^+} for 0i<n0 \leq i \lt n (equivalently, E iE i +E_i \supseteq E_{i^+} for 0i<n0 \leq i \lt n),
  • M 0M_0 consists of only isomorphisms/equivalences (equivalently, E 0E_0 consists of all morphisms), and
  • M nM_n consists of all morphisms (equivalently, E nE_n consists of only isomorphisms/equivalences).

(Or course, an nn-ary factorisation system is determined by the n1n - 1 factorisations systems (E i,M i)(E_i,M_i) for 0<i<n0 \lt i \lt n, but the the other two exist.) Do you agree?

Given an nn-ary factorisation system, the (co)image of (E i,M i)(E_i,M_i) is the ii-(co)image of the entire nn-ary factorisation system. (This agrees with the terminology in CatCat for n=3n = 3, or more generally with the terminology in (n2)Cat(n - 2) Cat or even (,n2)Cat(\infty,n - 2) Cat.)

Then extending this definition to lower values of nn, every category (or \infty-category) has a unique 11-ary factorisation system, where (E 0,M 0)(E_0,M_0) is (iso,all) and (E 1,M 1)(E_1,M_1) is (all,iso), as you suggested.

A category has a 00-ary factorisation system if and only if it is a groupoid, in which case (E 0,M 0)(E_0,M_0) is both (iso,all) and (all,iso) at once. In other words, rather than requiring every morphism to be a 00-ary composite on the nose, we require every morphism to be a 00-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) 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 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.
Created on July 27, 2010 06:02:34 by David Roberts (