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Stream: deprecated: topos theory

Topic: cocompactness contd.

Morgan Rogers (he/him) (Dec 08 2020 at 17:17):

A comorphism of sites $F: (\mathcal{C},J) \to (\mathcal{D},K)$ consists of a functor $F: \mathcal{C} \to \mathcal{D}$ which has the cover lifting property. Any such induces a geometric morphism $f: \mathbf{Sh}(\mathcal{C},J) \to \mathbf{Sh}(\mathcal{D},K)$ whose inverse image functor $f^*$ is defined by $P \mapsto \mathbf{a}_J(P \circ F)$. This is an extension of the definition on representables: the covering lifting property guarantees that for an object $D$ of $\mathcal{D}$, $f^*(\ell(D)) = \mathbf{a}_J(\mathrm{Hom}_{\mathcal{D}}(F(-),D))$. Have a look at @Olivia Caramello's preprint from this summer for lots of results on comorphisms of sites. As you can probably tell, the sheafification involved in this definition can be challenging to work with. If only there were a way to move that operation inside the brackets...

Given a site $(\mathcal{C},J)$, recall that the plus construction is a process by which one turns presheaves on $\mathcal{C}$ into $J$-separated presheaves, and then turns these into $J$-sheaves. Given a presheaf $P$ on $\mathcal{C}$, we obtain $P^+$ as
$P^+(C) = \mathrm{colim}_{R \in J(C)}\mathrm{Match}(R,P)$, where the latter collection is the collection of matching families for $P$ indexed by the morphisms in the $J$-covering sieve $R$. Assuming the sites are small-generated, in the special case $P = \mathrm{Hom}_{\mathcal{D}}(F(-),D)$, the conditions on a matching family mean that we can express $\mathrm{Match}(R,P)$ as $\mathrm{Hom}_{\mathbf{Sh}(\mathcal{D},K)}(\mathrm{colim}_{R}F(C_i),\ell(D))$, where that colimit is indexed by $R$, viewed as a subcategory of $\mathcal{C}/C$, and is taken in $\mathbf{Sh}(\mathcal{D},K)$ in case $\mathcal{D}$ lacks the required colimits. I'll write $F_R(C)$ for that colimit object. Thus $P^+(C) = \mathrm{colim}_{R \in J(C)}\mathrm{Hom}_{\mathbf{Sh}(\mathcal{D},K)}(F_R(C),\ell(D))$.

Now for the link with my earlier post: if $\ell(D)$ happens to be a strongly cocompact object, since the $J$-covering sieves form a filtered diagram, we get an isomorphism $\mathrm{Hom}_{\mathbf{Sh}(\mathcal{D},K)}(\lim_{R \in J(C)}F_R(C),\ell(D)) \cong \mathrm{colim}_{R \in J(C)}\mathrm{Hom}_{\mathbf{Sh}(\mathcal{D},K)}(F_R(C),\ell(D))$. This looks exciting: a way to re-express the inverse image of a geometric morphism coming from a comorphism of sites in an almost-representable form!

In the examples of Grothendieck toposes I've looked at so far, strongly cocompact objects are too rare for this to be useful. But I haven't looked at a lot of examples yet!

Jens Hemelaer (Dec 08 2020 at 17:54):

Interesting! I sometimes have difficulties with working with comorphisms of sites in practice, maybe this can help me.

The same kind of argument applies when you compute stalks of strongly cocompact objects. The stalk of a sheaf $\mathcal{F}$ is
$\mathrm{colim}_{i }\, \mathcal{F}(C_i)$,
if the point is given by a formal cofiltered limit $\mathrm{lim}_{i } \, C_i$.

If $\mathcal{F}$ is strongly cocompact, then
$\mathrm{colim}_i \, \mathcal{F}(C_i) = \mathrm{Hom}(\mathrm{lim}_i \, \mathbf{y}C_i, \mathcal{F})$.

In a topological space $X$ the points are formal intersections $\mathrm{lim}_i\, U_i$ for open sets $U_i$. Now suppose that $X$ is Hausdorff without isolated points. The stalk of a strongly cocompact sheaf at a point like that is then:
$\mathrm{colim}_i \, \mathcal{F}(U_i) = \mathrm{Hom}(\mathrm{lim}_i \, \mathbf{y}U_i, \mathcal{F}) = \mathrm{Hom}(\varnothing, \mathcal{F}) = 1$.

So $\mathcal{F}$ has a unique element in each stalk, which can only happen if $\mathcal{F}$ is the terminal object (maybe you looked at this example already).

Morgan Rogers (he/him) (Dec 08 2020 at 18:11):

I hadn't thought about Hausdorff spaces in particular, but it occurs to me that something must be off here, since $\mathbf{Set}$ is the topos of sheaves on the one-point space, and it's easy to check that the initial object is strongly cocompact there.

Reid Barton (Dec 08 2020 at 18:16):

Is it? An inverse limit of nonempty sets can be empty

Morgan Rogers (he/him) (Dec 08 2020 at 18:21):

I think a cofiltered limit of non-empty sets can't be empty.

Reid Barton (Dec 08 2020 at 18:22):

How about the intersection of the subsets $[n, \infty)$ of the reals?

Morgan Rogers (he/him) (Dec 08 2020 at 18:22):

But I can't remember my basis for this belief, now that it's been called into question :upside_down:

Morgan Rogers (he/him) (Dec 08 2020 at 18:23):

I guess I have to modify my stackexchange answer too, then, thanks for pointing that out!!

Reid Barton (Dec 08 2020 at 18:24):

Maybe it can't happen if all the maps are surjections

Jens Hemelaer (Dec 08 2020 at 18:38):

Ah sorry... my argument fails for discrete spaces. So Hausdorffness is not the right assumption. I'll edit what I wrote.

Morgan Rogers (he/him) (Dec 08 2020 at 19:58):

Reid Barton said:

Maybe it can't happen if all the maps are surjections

I need to properly check whether the non-empty finite spaces are strongly cocompact now haha