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

Proof

Define the support $supp(A) = A_{\le -1}$ of an object $A$ to be the image of the unique morphism $A\to 1$. That, is $A\to supp(A) \to 1$ is an eso-ff factorization. Since $supp(A)\to 1$ 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 $pos(K) \hookrightarrow K$ 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 $A^{\mathbf{2}} \to A\times A$. Since esos are stable under pullback, $A_1\;\rightrightarrows\; A$ is a homwise-discrete category, and it clearly has a functor from $ker(A)$, so it is a (1,2)-congruence. Let $A\to P$ be its quotient. By the classification of congruences, $P$ is posetal. And if we have any $f:A\to Q$ where $Q$ is posetal, then we have an induced functor $ker(A) \to ker(f)$. But $Q$ is posetal, so $ker(f)$ is a (1,2)-congruence, and thus $ker(A) \to ker(f)$ factors through a functor $A_1\to ker(f)$. This then equips $f$ with an action by the (1,2)-congruence $A_1\;\rightrightarrows\; A$, so that it descends to a map $P\to Q$. 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 $A\to 1$ 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 $K$, the inclusion $disc(K) \hookrightarrow K$ 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 $A^{\mathbf{2}} \to A\times A$; this can be constructed with countable unions in the usual way. Then $A_1\;\rightrightarrows\; A$ 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 $B Z$).

Theorem

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