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

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

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

In any (1,2)-exact 2-category $K$ the inclusion $\mathrm{pos}(K)\hookrightarrow K$ of the posetal objects has a left adjoint called the **(0,1)-truncation**.

Given $A$, define ${A}_{1}$ to be the (ff) image of ${A}^{2}\to A\times A$. Since esos are stable under pullback, ${A}_{1}\phantom{\rule{thickmathspace}{0ex}}\rightrightarrows \phantom{\rule{thickmathspace}{0ex}}A$ is a homwise-discrete category, and it clearly has a functor from $\mathrm{ker}(A)$, 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 $\mathrm{ker}(A)\to \mathrm{ker}(f)$. But $Q$ is posetal, so $\mathrm{ker}(f)$ is a (1,2)-congruence, and thus $\mathrm{ker}(A)\to \mathrm{ker}(f)$ factors through a functor ${A}_{1}\to \mathrm{ker}(f)$. This then equips $f$ with an action by the (1,2)-congruence ${A}_{1}\phantom{\rule{thickmathspace}{0ex}}\rightrightarrows \phantom{\rule{thickmathspace}{0ex}}A$, 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 $\mathrm{pos}(K)$.

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.

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

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

Given $A$, define ${A}_{1}$ to be the equivalence relation generated by the image of ${A}^{2}\to A\times A$; this can be constructed with countable unions in the usual way. Then ${A}_{1}\phantom{\rule{thickmathspace}{0ex}}\rightrightarrows \phantom{\rule{thickmathspace}{0ex}}A$ 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.

The groupoid reflection is the hardest and also requires infinitary structure. Note that the 2-pretopos $\mathrm{FinCat}$ does not admit groupoid reflections (the groupoid reflection of the “walking parallel pair of arrows” is $BZ$).

In any (2,1)-exact and countably-extensive 2-category $K$, the inclusion $\mathrm{gpd}(K)\hookrightarrow K$ 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)