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Stream: deprecated: topos theory

Topic: petit presheaf topoi

sarahzrf (May 20 2020 at 08:21):

so i worked out that a point of a presheaf topos $\operatorname{PSh}(X)$ over a locale X is a filter of opens (...that's correct, right?), so that includes all ordinary points, but also a wide array of other kinds of "place" or "location", like things that are infinitesimally displaced from an actual point, or just entire opens

sarahzrf (May 20 2020 at 08:21):

if we view $\operatorname{PSh}(X)$ as a petit topos, what kind of space should this mean??

sarahzrf (May 20 2020 at 08:22):

perhaps these should be new "generic points" sorta like how Spec k[x₁, x₂, x₃] has more points than k³?

sarahzrf (May 20 2020 at 08:24):

for the record, i was thinking about this because i wanted to take germs other than at actual points of a space, and i was trying to figure out how to reconcile that with the nlab's suggestion that a stalk is given by taking inverse image along the point in question

sarahzrf (May 20 2020 at 08:25):

but all you really need to take germs at a point of a space is the neighborhood filter induced by the point, of course, so you can take germs at infinity easily by just using the filter of neighborhoods of infinity, even if infinity is not a point

sarahzrf (May 20 2020 at 08:25):

so i was wondering if you could generalize from points of the topos to something like "filters on the opens of the base space"

sarahzrf (May 20 2020 at 08:26):

...and if it's a localic topos, i guess that is points—of the presheaf topos...?

sarahzrf (May 20 2020 at 08:26):

what's going on here?

Morgan Rogers (he/him) (May 20 2020 at 09:08):

Well the points of $\mathbf{PSh}(C)$ correspond to flat functors $C \to \mathbf{Set}$. When $C$ is the frame of opens of a space, such a functor has to preserve the terminal object, monos and meets, so indeed, the points of the presheaf topos correspond to ordinary filters of opens.

Jens Hemelaer (May 20 2020 at 10:37):

Geometrically, $\mathbf{PSh}(X)$ can be seen as a topological space (because it is localic and has enough points).
In other words, there is a topological space $Y$ such that $\mathbf{Sh}(Y) \simeq \mathbf{PSh}(X)$. You can make sure that $Y$ is a sober topological space, so that the elements of $Y$ are precisely the topos-theoretic points of $\mathbf{Sh}(Y)$. Then $Y$ is the space of filters of opens of $X$, with a suitable topology. There is an embedding of locales $X \subseteq Y$ corresponding to the inclusion of toposes $\mathbf{Sh}(X) \subseteq \mathbf{PSh}(X)$. The points of $X$ are then precisely the filters that are completely prime.

Jens Hemelaer (May 20 2020 at 10:43):

You can also take the topological space $Y'$ defined as follows. The points are indexed by the opens of $X$. So for each open $U$ of $X$ there is an element $y_U \in Y$. The open sets are of the form:
$\hat{U} = \{ y_V : V \subseteq U \}$
In other words, $\hat{U}$ contains $y_V$ if and only if $U$ contains $V$.
Then the open sets $\hat{U}$ define a topology on $Y'$. For this topology $\mathbf{Sh}(Y') = \mathbf{PSh}(X)$.

Most of the time, this space $Y'$ is not sober, so it is not really the same as $Y$ above. But
$\mathbf{Sh}(Y') \simeq \mathbf{Sh}(Y) \simeq \mathbf{PSh}(X)$.

sarahzrf (May 20 2020 at 13:11):

thanks!

sarahzrf (May 20 2020 at 13:11):

is there a good source that covers this kind of material?

sarahzrf (May 20 2020 at 13:16):

also, what's the suitable topology? :)
is it something like... an open for each original open, containing all of the filters that contain the original open?

Jens Hemelaer (May 20 2020 at 14:20):

sarahzrf said:

also, what's the suitable topology? :)
is it something like... an open for each original open, containing all of the filters that contain the original open?

Yes, exactly the topology generated by these opens :)
So for each open $U$ of $X$, you can define an open $\hat{U}$ of $Y$, which is by definition
$\hat{U} = \{ F \text{ filter } : U \in F \}$
One caveat: $U = \bigcup_{i} U_i$ does not imply that $\hat{U} = \bigcup_{i} \hat{U_i}$.

sarahzrf (May 20 2020 at 14:34):

ah, that makes some kind of sense

sarahzrf (May 20 2020 at 14:34):

seems like that should correspond somehow to how PSh(X) is generated under colimits by O(X), but those colimits need not correspond to the existing colimits

Jens Hemelaer (May 20 2020 at 14:34):

sarahzrf said:

is there a good source that covers this kind of material?

I think the keyword for stuff like this is "Stone duality".
Looking at filters gives a duality between posets and algebraic dcpo's.
Looking at prime filters gives a duality between bounded distributive lattices and spectral spaces.
Looking at completely prime filters gives a duality between spatial frames and sober topological spaces.

By looking at the filters on a locale, you're forgetting the additional structure on the locale and just thinking of it as a poset, and applying the duality then gives an "algebraic dcpo".

Stone duality is older than topos theory, so often books & articles discussing these duality theorems don't use topos theory at all. I personally prefer translating everything to topos theory. If you want to do that, then I suggest Caramello's "A topos-theoretic approach to Stone-type dualities". You can also take a look at my preprint "Grothendieck topologies on posets" :mischievous:

Jens Hemelaer (May 20 2020 at 14:39):

sarahzrf said:

seems like that should correspond somehow to how PSh(X) is generated under colimits by O(X), but those colimits need not correspond to the existing colimits

Yes exactly. If $U = \bigcup_i U_i$ then $\hat{U}$ and $\bigcup_i \hat{U}_i$ do not contain the same filters, but they do contain the same completely prime filters (by definition of completely prime). So if you restrict to the completely prime filters, then these two get identified.

So $\mathbf{PSh}(X)$ is the free cocompletion of $O(X)$, while $\mathbf{Sh}(X)$ is some "non-free cocompletion".

sarahzrf (May 20 2020 at 15:46):

@Jens Hemelaer so if you start with a discrete space, this is gonna be a superspace of the stone-čech compactification, right?

Jens Hemelaer (May 20 2020 at 15:50):

Right! The Stone-Čech compactification is the space of prime filters, and it's a subspace of the space of filters as above.

Jens Hemelaer (May 20 2020 at 15:51):

So for discrete spaces this space of filters is very complicated.

Jens Hemelaer (May 20 2020 at 15:58):

It's easier if you start with $\mathrm{Spec}(\mathbb{Z})$, or equivalently the set of prime numbers with the cofinite topology. Then the open sets are complements of finite sets of primes. You can then compute that the filters are indexed by sets of primes. So if $S$ is a set of primes, then the associated filter is:
$F_S = \{ U \subseteq \mathrm{Spec}(\mathbb{Z}) \text{ open } : S \subseteq U \}$.
A filter is a bit like a "distribution" or a "divisor" here.