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Stream: deprecated: topos theory

Topic: logical properties and topological properties

Matteo Capucci (he/him) (Feb 01 2021 at 14:22):

@Jules Hedges posted an interesting question on Twitter:
https://twitter.com/_julesh_/status/1356191289432281088?s=20
Does anybody know more about this topic? All I know about it is in my replies to that tweet, most importantly this is the only direct link between cohomological properties and internal logic that I know of.

I don’t care to actually learn topos theory, but I wish I had fast shortcut intuition for translating between what a space looks like and what the internal logic of its sheaf topos looks like… like, what is the theory of a donut? What formula detects compactness? Etc.

- julesh (@_julesh_)

Morgan Rogers (he/him) (Feb 01 2021 at 14:31):

Some properties, like de Morgan-ness have easy translations:

The topos $\mathrm{Sh}(X)$ of sheaves on a topological space $X$ is de Morgan iff $X$ is extremally disconnected, i.e. the closure of every open subset is open.

Morgan Rogers (he/him) (Feb 01 2021 at 14:38):

Neil Barton posted a similar question last year on the CT mailing list regarding how the properties of a category $C$ translate into properties of $\mathrm{PSh}(C)$; some answers are known. @Jens Hemelaer and I translated between monoid properties and topological properties in our recent paper.
But no one has worked on a comprehensive dictionary as far as I know. I have an ambition to do this once I've accumulated enough general results regarding geometric morphisms, since it would be far more interesting to give a dictionary that makes sense over an arbitrary base topos rather than just $\mathrm{Set}$.

Fawzi Hreiki (Feb 01 2021 at 16:10):

Well there's Ingo Blechschmidt's thesis but thats more oriented towards AG

Peter Arndt (Feb 01 2021 at 18:30):

The axiom of choice fails for sheaves over every non-discrete $T_1$-space: AC implies Booleanness (Moerdijk/MacLane, chapter VI, exercise 16), i.e. that the lattice of open sets is a Boolean algebra; for a $T_1$-space this means that it is discrete (Moerdijk/MacLane, ch.VI, ex. 3)

Matteo Capucci (he/him) (Feb 02 2021 at 09:09):

Fawzi Hreiki said:

Well there's Ingo Blechschmidt's thesis but thats more oriented towards AG

That's how to use internal language to do AG, it's a different thing.
Of course there are some results (already in the literature) about how topological properties of a space are reflected in the internal logic of a space, like quasi-compactness and EP, booleannes and separation, and so on.

Matteo Capucci (he/him) (Feb 02 2021 at 09:11):

What I'm interested here is how cohomological/homotopical properties of a space (say, a manifold) are reflected in the internal logic.

Matteo Capucci (he/him) (Feb 02 2021 at 09:13):

Let's use $S^1$ as a toy example. In this case, there can be locally constant sheaves with no global sections, which testify the presence of a non-trivial cocycle in $H^1(S^1, A)$ for $A$ abelian.
Global sections are definitely an internalizable property: they are elements of the terminal type. So one might conjecture that, internally, $H^1$ is reflected in the kind of types without global elements, hence in the 'failure of well-pointedness'.

Matteo Capucci (he/him) (Feb 02 2021 at 09:15):

THIS IS FALSE for trivial reasons:

One easy observation is that if $H^n(X,G) = 0$ for every $n$, then $X$ is weakly equivalent to a point hence its topos of sheaves is (right?) equivalent to $\mathbf{Set}$

Jens Hemelaer (Feb 02 2021 at 09:41):

Matteo Capucci (he/him) said:

One easy observation is that if $H^n(X,G) = 0$ for every $n$, then $X$ is weakly equivalent to a point hence its topos of sheaves is (right?) equivalent to $\mathbf{Set}$

If $X$ is Hausdorff then it can be completely recovered from the topos of sheaves on it, so in particular the topos of sheaves is not homotopy invariant.

Something closely related is the case though: if $X$ is contractible, then every geometric morphism $\mathbf{Sh}(X) \to \mathbf{PSh}(G)$, where $G$ is some discrete group, factorizes through $\mathbf{Set}$. I don't know how to translate this to a logical property.

Jens Hemelaer (Feb 02 2021 at 12:30):

There seem to be multiple possible interpretations for "a property holds in the internal logic of a certain topos".
For example, a topos is said to satisfy De Morgan's law if this law holds for subobjects of any object in the topos.
On the other hand, in Ingo Blechschmidt's thesis, a property is said to be satisfied by the topos if it is satisfied by subobjects of the terminal object.

If you follow the first interpretation, then all properties satisfied by $\mathbf{Sh}(X)$ will also be satisfied by $\mathbf{Sh}(Y)$, whenever there is a local homeomorphism $Y \to X$. So there can be no property that expresses (in this sense) compactness or nontrivial cohomology.

On the other hand, as @Matteo Capucci (he/him) already pointed out, there is a logical characterization of compactness if you look at subobjects of the terminal object.

Then there are properties like AC that are statements involving types/objects, so maybe there the ambiguity disappears.

John Baez (Feb 02 2021 at 17:54):

I'm confused by your "easy observation", Matteo: [0,1] with its usual topology is contractible so all its cohomology groups are trivial, but the topos of sheaves on [0,1] isn't equivalent to $\mathsf{Set}$, is it?

John Baez (Feb 02 2021 at 17:56):

Oh, okay, Jens already tackled this problem in a lot more detail.

John Baez (Feb 02 2021 at 18:01):

By the way, Matteo, on a much more minor note: it's not standard in algebraic topology to use $G$ as notation for an abelian group, as in $H^n(X,G)$. Algebraic topologists always use $A$ for an abelian group. When I see $H^n(X,G)$ I instinctively think wow, he's considering nonabelian cohomology! - which exists, but is quite tricky. $H^n(X,A)$, on the other hand, is a familiar friend.

Matteo Capucci (he/him) (Feb 03 2021 at 09:14):

Jens Hemelaer said:

There seem to be multiple possible interpretations for "a property holds in the internal logic of a certain topos".
For example, a topos is said to satisfy De Morgan's law if this law holds for subobjects of any object in the topos.
On the other hand, in Ingo Blechschmidt's thesis, a property is said to be satisfied by the topos if it is satisfied by subobjects of the terminal object.

Uh -- I thought the two things were equivalent, aren't they?

Matteo Capucci (he/him) (Feb 03 2021 at 09:16):

John Baez said:

By the way, Matteo, on a much more minor note: it's not standard in algebraic topology to use $G$ as notation for an abelian group, as in $H^n(X,G)$. Algebraic topologists always use $A$ for an abelian group. When I see $H^n(X,G)$ I instinctively think wow, he's considering nonabelian cohomology! - which exists, but is quite tricky. $H^n(X,A)$, on the other hand, is a familiar friend.

Fair enough :) All my naivety and ignorance is showing in this thread :laughing:

Morgan Rogers (he/him) (Feb 03 2021 at 09:34):

Matteo Capucci (he/him) said:

Jens Hemelaer said:

There seem to be multiple possible interpretations for "a property holds in the internal logic of a certain topos".

Uh -- I thought the two things were equivalent, aren't they?

The subterminal objects in any topos correspond to the global sections of the subobject classifier, and the subsubterminal objects correspond to the morphisms from subterminal objects to the subobject classifier :stuck_out_tongue_wink: . The resulting Heyting algebras of subobjects inherit/externalise the internal properties of $\Omega$ as an internal Heyting algebra; for example, they will be Boolean/de Morgan algebras externally if $\Omega$ is one of these internally. In general, looking at the algebras of subobjects of subterminal objects will not be enough to determine the internal logic, though, since these algebras could have extra properties. In a topos of presheaves on a monoid, for example, these are always the two-element and degenerate Boolean algebra (any such topos is two-valued), but most such toposes are far from Boolean!
In a localic topos, however, (in particular, in $\mathrm{Sh}(X)$ for any space $X$), the subterminal objects generate, so that the algebras of subobjects of subterminal objects are enough to determine the properties of the subobject classifier as an internal Heyting algebra. In this case, the two things should be equivalent, but I expect there are some caveats regarding what kinds of property you are considering.

Jens Hemelaer (Feb 03 2021 at 11:48):

Matteo Capucci (he/him) said:

Jens Hemelaer said:

There seem to be multiple possible interpretations for "a property holds in the internal logic of a certain topos".
For example, a topos is said to satisfy De Morgan's law if this law holds for subobjects of any object in the topos.
On the other hand, in Ingo Blechschmidt's thesis, a property is said to be satisfied by the topos if it is satisfied by subobjects of the terminal object.

Uh -- I thought the two things were equivalent, aren't they?

Yes, they are supposed to be equivalent, and in the topological space case this is Proposition 2.4 in Blechschmidt's thesis.
The problem is that then global properties such as compactness and cohomology can't be formulated as properties of the internal logic.
Blechschmidt solves this by talking about "metaproperties" instead. Using metaproperties you can express non-local properties such as compactness and probably cohomology.

So "what does the internal logic of a topos look like" depends on whether or not you include metaproperties. If you don't, then any manifold of dimension $n$ will have the same internal logic as $\mathbb{R}^n$ (for example, none of them satisfy De Morgan's law, and none of them satisfy LEM).

Matteo Capucci (he/him) (Feb 03 2021 at 14:21):

Ooh I see the confusion. Indeed, I'm talking about metaproperties, otherwise, as you point out, there's no hope to see global structure.

John Baez (Feb 03 2021 at 16:06):

Jens Hemelaer said:

So "what does the internal logic of a topos look like" depends on whether or not you include metaproperties. If you don't, then any manifold of dimension $n$ will have the same internal logic as $\mathbb{R}^n$ (for example, none of them satisfy De Morgan's law, and none of them satisfy LEM).

Do people know any facts about the internal logic of $\mathbb{R}^n$ that depend on $n$ in an interesting way? (Not interesting: "if $n \gt 0$ then...")

Jens Hemelaer (Feb 04 2021 at 17:21):

I would be interested in an answer to @John Baez's question as well...
These are the axioms for which I checked the literature: law of excluded middle, De Morgan's law, the Gödel–Dummett axiom and the Kreisel–Putnam axiom (for the last two, see here). Unfortunately, none of them depend on $n$ in an interesting way.

Morgan Rogers (he/him) (Feb 04 2021 at 17:25):

Are there any nice descriptions of the differences between the frames of opens between, say, $\mathbb{R}$ and $\mathbb{R}^2$? That's already a hard distinction to articulate, right?

John Baez (Feb 04 2021 at 17:44):

Is there any way to detect, from the category of sheaves on a space $X$, the local homology groups $H_n(X, X- \{x\})$ or local cohomology groups $H^n(X, X- \{x\})$?

John Baez (Feb 04 2021 at 17:44):

That would serve to distinguish $\mathbb{R}^k$'s for different $k$.

Morgan Rogers (he/him) (Feb 04 2021 at 17:49):

Considering local complements of points feels weird, but that data is intrinsic to the spaces, so it should in principal be extractable... That's a lot of machinery to plop on top, but it would be interesting to see what the logical translations of these things is!!

John Baez (Feb 04 2021 at 18:36):

Local complements of points are really important in algebraic topology: intuitively, $H^n(X, X- \{x\})$ reveals the extent to which poking a hole at $x$ reveals a little $n$-sphere surrounding that point.

• What exactly are the elements of a local homology group?

John Baez (Feb 04 2021 at 18:38):

If $H^1(X, X - \{x\})$, moving something around a little loop around $x$ can do something nontrivial, i.e. there are nontrivial covering spaces of a little 1-sphere surrounding $x$. Since covering spaces are pretty easy to think about in terms of sheaves, I think this case may be the easiest.

John Baez (Feb 04 2021 at 18:39):

And it may have a "logical" meaning, sort of like "if we remove this situation from consideration, walking around this situation we may find that names of things get interchanged". (That's a pretty vague description that I'm hoping a topos theorist could make precise.)

Morgan Rogers (he/him) (Feb 05 2021 at 11:44):

I'm familiar with the algebraic topology reasons for considering the $H^n(X,X - \{x\})$, and it's the simplest construction I know of that clearly differentiates the $\mathbb{R}^n$. I was trying to point out that points aren't primitive in toposes, so $H^n(X,X - \{x\})$ starts to feel contrived. Now that I remember that the relative cohomology groups $H^n(X,U)$ are defined for any open subset $U$, I can see that these are members of a class of invariants that can be discussed purely in terms of open sets, which will make things easier!

David Michael Roberts (Feb 05 2021 at 11:59):

The data of $X$, $x$ and $X - \{x\}$ reminds me of a recollement-type situation (https://ncatlab.org/nlab/show/recollement), or constructible/perverse sheaves or something. Where people push forward a sheaf along the inclusion of $\{x\}$, into $X$ and then pull the result back to $X - \{x\}$ . Just a random observation.

Aha:

Intersection cohomology, unlike usual cohomology, can distinguish between spaces that are homotopic (but not homeomorphic) to one another. For example, as we shall see, even though every point in X has a contractible neighborhood, the local intersection cohomology and the stalks [...] need not be trivial. [see section 4 of arXiv:math/0307349]