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 K the inclusion Sub(1)K of the subterminal objects has a left adjoint called the support or (-1)-truncation.

Proof

Define the support supp(A)=A 1 of an object A to be the image of the unique morphism A1. That, is Asupp(A)1 is an eso-ff factorization. Since supp(A)1 is ff, supp(A) is subterminal, and since esos are orthogonal to ffs, it is a reflection into Sub(1).

(0,1)-truncation

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

Theorem

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

Proof

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

This is actually a special case of the (eso+full,faithful) factorization system?, since an object A is posetal iff A1 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 K, the inclusion disc(K)K of the discrete objects has a left adjoint called the 0-truncation or discretization.

Proof

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

There are other sufficient conditions on K 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 FinCat does not admit groupoid reflections (the groupoid reflection of the “walking parallel pair of arrows” is BZ).

Theorem

In any (2,1)-exact and countably-extensive 2-category K, the inclusion gpd(K)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)