Michael Shulman
truncation in an exact 2-category

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

(-1)-truncation

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

Theorem

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.

Proof

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).

(0,1)-truncation

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

Theorem

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.

Proof

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.

0-truncation

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

Theorem

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.

Proof

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.

(1,0)-truncation

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).

Theorem

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? (129.241.15.200)