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 XOb(C) let C[X] be the category given by the pullback codisc(X)× codisc(Obj(C))C (In other words, objects X and arrows X 2× Obj(C) 2Mor(C)). Then for a functor f:DC there is a factorisation

DC[Obj(D)]C[im(f 0)]CD \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 =(X 1X 0) with composition (something like a (pointed magma)-oid :) and a map of such things X C 1 for the underlying 1-category C 1 of a 2-category (this works for bicategories too, but let’s stick to strict things). In particular, for the underlying category D 1 of a second 2-category D equipped with a 2-functor F:DC. In general this is a bicategory, but if X is a category, then C[X ] 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[?] 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(n+1)Cat adding to each category a unique arrow between any two parallel n-arrows. For an n-functor DC define C[D m]=codisc m+1(D m)× codisc(C 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,,n1. 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 compBicat 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 CatrGrp comp lands in 2CatBicat.

Comment 25 (Toby Bartels):

For n>1, I claim that an n-ary factorisation system consists of n + (that is n+1) factorisation systems (E i,M i) (for 0in) such that

  • M iM i + for 0i<n (equivalently, E iE i + for 0i<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 n1 factorisations systems (E i,M i) for 0<i<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 (n2)Cat or even (,n2)Cat.)

Then extending this definition to lower values of n, every category (or -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) α. Then an α-stage factorisation system (in an ambient -category C) consists of an α-indexed family of factorisation systems in C such that:

  • M iM j whenever ij (equivalently, E iE j whenever ij),
  • each morphism f:XY is both the inverse limit limiim if in the slice category C/Y and the direct limit colimicoim if in the coslice category X/C, and
  • for each f:XY, id Y is colimiim if and id X is limicoim if.
Created on July 27, 2010 06:02:34 by David Roberts (