Michael Shulman
truncation in an exact 2-category

In a suitably exact 2-category, we can construct truncations as quotients of suitable congruences.


This case is easy and just like for 1-categories.


In any regular 2-category KK the inclusion Sub(1)KSub(1) \hookrightarrow K of the subterminal objects has a left adjoint called the support or (-1)-truncation.


Define the support supp(A)=A 1supp(A) = A_{\le -1} of an object AA to be the image of the unique morphism A1A\to 1. That, is Asupp(A)1A\to supp(A) \to 1 is an eso-ff factorization. Since supp(A)1supp(A)\to 1 is ff, supp(A)supp(A) is subterminal, and since esos are orthogonal to ffs, it is a reflection into Sub(1)Sub(1).


Perhaps surprisingly, the next easiest case is the posetal reflection.


In any (1,2)-exact 2-category KK the inclusion pos(K)Kpos(K) \hookrightarrow K of the posetal objects has a left adjoint called the (0,1)-truncation.


Given AA, define A 1A_1 to be the (ff) image of A 2A×AA^{\mathbf{2}} \to A\times A. Since esos are stable under pullback, A 1AA_1\;\rightrightarrows\; A is a homwise-discrete category, and it clearly has a functor from ker(A)ker(A), so it is a (1,2)-congruence. Let APA\to P be its quotient. By the classification of congruences, PP is posetal. And if we have any f:AQf:A\to Q where QQ is posetal, then we have an induced functor ker(A)ker(f)ker(A) \to ker(f). But QQ is posetal, so ker(f)ker(f) is a (1,2)-congruence, and thus ker(A)ker(f)ker(A) \to ker(f) factors through a functor A 1ker(f)A_1\to ker(f). This then equips ff with an action by the (1,2)-congruence A 1AA_1\;\rightrightarrows\; A, so that it descends to a map PQP\to Q. It is easy to check that 2-cells also descend, so PP is a reflection of AA into pos(K)pos(K).

This is actually a special case of the (eso+full,faithful) factorization system?, since an object AA is posetal iff A1A\to 1 is faithful. The proof is also an evident specialization of that.


The discrete reflection, on the other hand, requires some additional structure.


In any 1-exact and countably-coherent 2-category KK, the inclusion disc(K)Kdisc(K) \hookrightarrow K of the discrete objects has a left adjoint called the 0-truncation or discretization.


Given AA, define A 1A_1 to be the equivalence relation generated by the image of A 2A×AA^{\mathbf{2}} \to A\times A; this can be constructed with countable unions in the usual way. Then A 1AA_1\;\rightrightarrows\; A is a 1-congruence, and as in the posetal case we can show that its quotient is a discrete reflection of AA.

There are other sufficient conditions on KK for the discretization to exist; see for instance classifying cosieve. We can also derive it if we have groupoid reflections, since the discretization is the groupoid reflection of the posetal reflection.


The groupoid reflection is the hardest and also requires infinitary structure. Note that the 2-pretopos FinCatFinCat does not admit groupoid reflections (the groupoid reflection of the “walking parallel pair of arrows” is BZB Z).


In any (2,1)-exact and countably-extensive 2-category KK, the inclusion gpd(K)Kgpd(K) \hookrightarrow K of the groupoidal objects has a left adjoint called the (1,0)-truncation.

Revised on May 29, 2012 22:04:00 by Andrew Stacey? (