Every Grothendieck topos can be built from localic topoi

Question

Theorem 2 in these notes[1] states that, roughly, that each Grothendieck topos can be built (using limits and colimits) from localic topoi. To what extent is that related to the theorem of Joyal and Tierney which states that each Grothendieck topos is equivalent to the topos of equivariant sheaves on a groupoid in the category of locales?

1. Jacob Lurie, 2018, lecture notes from Math 278X Categorical Logic, https://www.math.ias.edu/~lurie/278x.html, Lecture 16 Enumerations

Answer 1

The groupoid representation of Joyal and Tierney may be identified with the truncated simplicial diagram appearing in the statement of Theorem 2 of the Lurie notes mentioned in the question. That is, a groupoid object is precisely a diagram of the shape indicated in Theorem 2 satisfying some axioms, which are automatically satisfied by construction when one takes pseudopullbacks as indicated in the diagram. In Joyal and Tierney's theorem, the groupoid in localic toposes is precisely this groupoid: it has topos of objects $U$, topos of 1-cells $U \times_X U$, and so forth.

To elaborate a bit, the diagram displayed in Theorem 2, which looks like

$U \times_X U \times_X U ^\to_\to \to U \times_X U ^\to_\to U$

(implicitly there are also some maps going back in the reverse direction) is really just the first few stages of the Cech nerve of the map $U \to X$. The Cech nerve as a whole is indexed by the dual simplex category $\Delta^{op}$, and here we just have the part on the 3 smallest objects of $\Delta^{op}$, namely $[0], [1], [2]$. It may be more familiar that when you truncate all the way to just two objects $[0],[1]$, you get the maps $U \times_X U ^\to_\to U$ -- the kernel pair of the map $U \to X$, which is an equivalence relation. The higher analog of the fact that a kernel pair is always an equivalence relation is that the Cech nerve of a map is always a groupoid object -- $[0]$ corresponds to the objects of the groupoid, $[1]$ corresponds to the morphisms, $[2]$ corresponds to composable pairs of morphisms (not to 2-cells), and in general $[n]$ corresponds to composable $n$-tuples of morphisms.

You can more generally think of an internal category object (not just an internal groupoid object) in a category as a certain type of simplicial object, as described here.

Answer 2

They are (it is?) the same theorem, but emphasising different aspects.

We can exploit the object classifier to get from the formulation in terms of (pseudo)colimits to the "elementary" formulation in terms of equivariant sheaves. The point is that, for every Grothendieck topos $\mathcal{E}$, the functor $\textbf{Geom} (\mathcal{E}, [\textbf{Set}_\textrm{fin}, \textbf{Set}]) \to \mathcal{E}$ sending a geometric morphism $f : \mathcal{E} \to [\textbf{Set}_\textrm{fin}, \textbf{Set}]$ to the object $f^* A$ in $\mathcal{E}$, where $A : \textbf{Set}_\textrm{fin} \to \textbf{Set}$ is the inclusion, is fully faithful and essentially surjective on objects. That is, $[\textbf{Set}_\textrm{fin}, \textbf{Set}]$ represents the contravariant forgetful 2-functor that sends a Grothendieck topos to its underlying category. It immediately follows that (bi)colimits in the 2-category of Grothendieck toposes are (bi)limits of the underlying categories. (It is awkward to keep using the 2- / bi- prefixes, so I will omit it from here onwards.)

So, for example, the initial Grothendieck topos is the terminal category, the coproduct of Grothendieck toposes is the product of the underlying categories, the pushout of Grothendieck toposes is the pullback of the underlying categories, etc. What about the colimit of the diagram appearing in Lurie's formulation of the theorem? Well, the underlying category is the limit of the diagram of the underlying categories, and because the forgetful functor is contravariant, the shape of the diagram is also dualised. If you write out explicitly the universal property of the limit, you will find that it is an extremely terse way of writing something that looks like a groupoid action.

Indeed, suppose given categories $\mathcal{C}^0$, $\mathcal{C}^1$, and $\mathcal{C}^2$ and functors $d^0, d^1 : \mathcal{C}^0 \to \mathcal{C}^1$, $d^0, d^1, d^2 : \mathcal{C}^1 \to \mathcal{C}^2$, satisfying the cosimplicial identities (strictly, to preserve our sanity). So we have a truncated cosimplicial object in $\textbf{Cat}$, and its limit is (up to equivalence!) the following category:

• An object consists of the following data:

  • An object $X^i$ in each $\mathcal{C}^i$.
  • An isomorphism $d^j X^i \cong X^{i+1}$ for each $i$ and $j$, such that for all $j$ and $k$, the composite $$\require{AMScd} \begin{CD} d^k d^j X^0 @>{d^k (\cong)}>> d^k X^1 @>{\cong}>> X^2 \end{CD}$$ depends only on the composite $d^k d^j$ in the simplex category $\mathbf{\Delta}$. So, for example, because $d^2 d^0 = d^0 d^1$ in $\mathbf{\Delta}$, the following diagram in $\mathcal{C}^2$ commutes: $$\begin{CD} d^0 d^1 X^0 @>{d^0 (\cong)}>> d^0 X^1 @>{\cong}>> X^2 \\ @| && @| \\ d^2 d^0 X^0 @>>{d^2 (\cong)}> d^2 X^1 @>>{\cong}> X^2 \end{CD}$$

• A morphism consists of a morphism in each $\mathcal{C}^i$, compatible with the structural isomorphisms in the obvious way.

• Composition and identities are inherited from each $\mathcal{C}^i$.

Manipulating the structural isomorphisms and eliminating the objects $X^1$ and $X^2$ gives us an isomorphism $d^1 X^0 \cong d^0 X^0$ in $\mathcal{C}^1$ and the following commutative diagram in $\mathcal{C}^2$: $$\begin{CD} d^2 d^1 X^0 @= d^1 d^1 X^0 @>{d^1 (\cong)}>> d^1 d^0 X^0 \\ @V{d^2 (\cong)}VV && @| \\ d^2 d^0 X^0 @= d^0 d^1 X^0 @>>{d^0 (\cong)}> d^0 d^0 X^0 \end{CD}$$ The isomorphism $d^1 X^0 \cong d^0 X^0$ in $\mathcal{C}^1$ can be thought of as the groupoid action on $X^0$, and the commutative diagram in $\mathcal{C}^2$ can be thought of as the compatibility between the groupoid action on $X^0$ and composition in the groupoid itself.

If this still seems vague, then take the case where we have a groupoid $\mathcal{G}$ and $\mathcal{C}^i = \textbf{Set}^{G_i}$, where $G_0$ is the set of objects of $\mathcal{G}$, $G_1$ is the set of morphisms of $\mathcal{G}$, and $G_2$ is the set of composable pairs of morphisms of $\mathcal{G}$, and each of the functors $d^j : \mathcal{C}^i \to \mathcal{C}^{i+1}$ are defined by reindexing families of sets in the following way:

• $d^0 : \mathcal{C}^0 \to \mathcal{C}^1$ sends each $G_0$-indexed family of sets to the $G_1$-indexed family where the $g$-th component is the component of the original family at the codomain of $g$.

• $d^1 : \mathcal{C}^0 \to \mathcal{C}^1$ is defined similarly, except for using the domain of $g$ instead of the codomain of $g$.

• $d^0 : \mathcal{C}^1 \to \mathcal{C}^2$ sends each $G_1$-indexed family of sets to the $G_2$-indexed family where the $(h, g)$-th component is the $h$-th component of the original family.

• $d^1 : \mathcal{C}^1 \to \mathcal{C}^2$ sends each $G_1$-indexed family of sets to the $G_2$-indexed family where the $(h, g)$-th component is the $h \circ g$-th component of the original family.

• $d^2 : \mathcal{C}^1 \to \mathcal{C}^2$ sends each $G_1$-indexed family of sets to the $G_2$-indexed family where the $(h, g)$-th component is the $g$-th component of the original family.

It is straightforward, if extremely tedious, to verify that the limit category is equivalent to $[\mathcal{G}, \textbf{Set}]$. The definition of an action of a localic/topological groupoid is based on this observation, replacing $\textbf{Set}^{G_i}$ with $\textbf{Sh} (G_i)$.

I do not think there is an abstract nonsense argument to get back from the "elementary" formulation to the colimit formulation. If the colimit of the diagram of Grothendieck toposes exists then its underlying category must be the limit, but a priori we do not even know that the limit category is a Grothendieck topos, let alone one that has the required universal property. So the colimit formulation is stronger in some sense.