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Stream: deprecated: topos theory

Topic: A 2-category of pointed toposes?

Keith Elliott Peterson (Aug 10 2021 at 05:31):

I'm just getting started understanding toposes, but I was wondering: If we can define toposes that model sets over spaces and/or time (or possibly some other notion of variation), is it possible to pick out a specific place and/or moment of variation, much like how in the category of pointed sets, we specify a basepoint for each set?

From my understanding, points of a topos are the geometric morphisms out of Set, so it doesn't seem too farfetched to come up with a category of pointed toposes. Are there any interesting properties or pitfalls of such I should be aware of?

fosco (Aug 10 2021 at 08:17):

What is your definition of a pointed topos? If you mean a topos which is also a pointed category, this is not possible: a topos is cartesian closed, a pointed category has a zero object. These two classes of categories intersect trivially, because if $\cal C$ has a zero object $0\cong 1$ then $A\cong A\times 1 \cong A \times 0 \cong 0$, and thus $\cal C$ is the terminal category.

Note that something similar happens also if you want to give each presheaf $F: {\cal C} \to {\rm Set}$ a distinguished point; the resulting category has a zero object, so it can't be a (nontrivial) topos.

Morgan Rogers (he/him) (Aug 10 2021 at 09:01):

Taking a more positive approach, there are a few things which you could mean that will produce meaningful nontrivial categories.

Taking what you said at face value, there is a 2-category of "Grothendieck toposes equipped with a point (geometric morphism from Set)", $\mathrm{Set}/\mathfrak{BTop}$, which includes the examples you mentioned. For example, for each real number $r \in \mathbb{R}$ there is a pointed topos $\mathrm{Sh}(\mathbb{R})_r$ obtained by equipping the topos of sheaves on $\mathbb{R}$ with the corresponding point.
This 2-category of pointed toposes is not well-studied, but has some nice properties. Most notably for me is the fact that toposes of monoid and group actions (discrete or topological!) come with a distinguished point and are coreflective in this 2-category: for any pointed topos you can obtain a monoid or group by taking the endo/automorphisms of the distinguished point, and the canonical point of the corresponding topos of actions will factor that distinguished point.

Alternatively, closer to what @fosco was saying: Given a topos $\mathcal{E}$ you can consider the category of pointed objects of the topos, which form the coslice category $1/\mathcal{E}$; there is a geometric (indeed, essentially algebraic) theory of pointed objects, with classifying topos $[\mathrm{FinSet}_*,\mathrm{Set}]$, where $\mathrm{FinSet}_*$ is the category of finite pointed sets, so $1/\mathcal{E} \simeq \mathrm{Geom}(\mathcal{E},[\mathrm{FinSet}_*,\mathrm{Set}])$.

Mike Shulman (Aug 10 2021 at 13:11):

The 2-category $\mathrm{Set}/\mathfrak{BTop}$ of pointed toposes appears in some places. For instance, there are notions of "localization" of a topos at a point, discussed somewhere in part C of the Elephant.

Morgan Rogers (he/him) (Aug 10 2021 at 13:34):

Good point! That will be in C3.6, where local and totally connected geometric morphisms are discussed.

Keith Elliott Peterson (Aug 11 2021 at 06:52):

I don't own any of the books of The Elephant, but yes, I wish to consider Grothendieck toposes equipped with a point.

fosco (Aug 11 2021 at 09:08):

Keith Peterson said:

I don't own any of the books of The Elephant, but yes, I wish to consider Grothendieck toposes equipped with a point.

good! Yes, I didn't mention that "an object in $Set/BTop$" was a more promising interpretation of your request. Sorry if it meant too destructive a comment!

Keith Elliott Peterson (Aug 11 2021 at 20:41):

fosco said:

Keith Peterson said:

I don't own any of the books of The Elephant, but yes, I wish to consider Grothendieck toposes equipped with a point.

good! Yes, I didn't mention that "an object in $Set/BTop$" was a more promising interpretation of your request. Sorry if it meant too destructive a comment!

It's fine. Miscommunication is bound to happen since I'm learning the language of toposes.

Morgan Rogers (he/him) (Aug 12 2021 at 13:12):

Is there anything you want to know in particular about this 2-category?

Keith Elliott Peterson (Aug 12 2021 at 23:09):

Actually, are there any difficulties in defining objects in this 2-category? I've never localized in any topos, let alone at a point.

John Baez (Aug 12 2021 at 23:30):

I thought an object of this 2-category was a topos $X$ equipped with a "point", meaning a geometric morphism $X \to \mathrm{Set}$.

John Baez (Aug 12 2021 at 23:30):

Have I lost track of the plot?

Mike Shulman (Aug 13 2021 at 03:14):

You don't need to localize to get an object of this category. Sheaves on any pointed topological space is a pointed topos, for instance. Localization is an extra thing that you can do to a pointed topos.

Keith Elliott Peterson (Aug 13 2021 at 07:38):

Mike Shulman said:

You don't need to localize to get an object of this category. Sheaves on any pointed topological space is a pointed topos, for instance. Localization is an extra thing that you can do to a pointed topos.

Thanks for the clarification. To get further clarity, in such a pointed topos, we can localize at the chosen basepoint, yes?

Keith Elliott Peterson (Aug 13 2021 at 08:10):

John Baez said:

I thought an object of this 2-category was a topos $X$ equipped with a "point", meaning a geometric morphism $X \to \mathrm{Set}$.

The nlab gives points in toposes in the opposite direction.

My limited understanding is that $\mathrm{Set}$ is terminal in the 2-category of Grothendieck toposes, so a (generalized) point would be a morphism out of this terminal object.

Fawzi Hreiki (Aug 13 2021 at 08:53):

John Baez said:

I thought an object of this 2-category was a topos $X$ equipped with a "point", meaning a geometric morphism $X \to \mathrm{Set}$.

Geometric morphisms are 'flipped' the same way maps of locales are reversed homomorphisms of frames. I've seen maps in the other direction (i.e. lex left adjoints) called 'algebraic morphisms' I guess in analogy to the frame-locale duality.

Fawzi Hreiki (Aug 13 2021 at 08:55):

Just for completeness sake: a geometric morphism is a functor (direct image) with a lex left adjoint (inverse image). The directionality can definitely be confusing at times.

David Michael Roberts (Aug 13 2021 at 10:35):

@Keith Peterson yes, a point of a topos is a geometric morphism $Set \to X$, which is the same as asking for just a left adjoint functor $X\to Set$ that preserves finite limits, which for $X$ a Grothendieck topos is the same as just asking fo a finite limit preserving cocontinuous functor $X\to Set$ (the adjoint functor theorem then tells us this has a right adjoint).

Mike Shulman (Aug 13 2021 at 15:19):

This confusion is a good reason to switch notation when we pass to the opposite category. In the case of locales, we say that a locale $X$ "is" a frame $\mathcal{O}(X)$, and a continuous map of locales $f:X\to Y$ is a frame homomorphism $f^*:\mathcal{O}(Y)\to \mathcal{O}(X)$. Similarly, some people say that a topos $X$ "is" a "Giraud frame" or "logos" $\mathcal{O}(X)$ (meaning the category of sheaves on some small site), and a geometric morphism $f:X\to Y$ is a logos morphism $f^* : \mathcal{O}(Y) \to \mathcal{O}(X)$ (meaning a left exact left adjoint).

Mike Shulman (Aug 13 2021 at 15:20):

Then the terminal topos $\ast$ is defined by $\mathcal{O}(\ast) = \rm Set$, and so it is unambiguous to talk about a geometric morphism $\ast \to X$ which corresponds to a logos map $\mathcal{O}(X) \to \rm Set$.

Mike Shulman (Aug 13 2021 at 15:21):

But regardless of whether we call the terminal topos $\ast$ or $\rm Set$, it's pretty standard that when we use the word "geometric morphism" we mean the "geometric direction". So even if we use the same notation for toposes and their underlying logoses, there is a unique geometric morphism $X\to \rm Set$, while a point of a topos is a geometric morphism $\rm Set \to X$.

Keith Elliott Peterson (Aug 14 2021 at 03:38):

I changed the topic title to better reflect the topic at hand.

Also, just as there is a 2-functor,

$$[-,\mathrm{Set}]: \mathrm{Cat} \to \mathfrak{BTop} ,$$

taking categories to their category of (co)presheaves, does it also hold there is an analogous 2-functor,

$$[-,\mathrm{Set}]: \mathrm{Cat}|_\text{pointed} \to\mathrm{Set}/\mathfrak{BTop},$$

taking pointed categories to their category of "pointed" (co)presheaves?

John Baez (Aug 14 2021 at 04:06):

Keith Peterson said:

John Baez said:

I thought an object of this 2-category was a topos $X$ equipped with a "point", meaning a geometric morphism $X \to \mathrm{Set}$.

The nlab gives points in toposes in the opposite direction.

Okay, you're right.

John Baez (Aug 14 2021 at 04:09):

A geometric morphism is a pair of adjoint functors, with the left adjoint also preserving finite limits.

John Baez (Aug 14 2021 at 04:10):

So, it has arrows going both ways. One is primary if you're doing geometry, the other is primary if you're doing algebra.

John Baez (Aug 14 2021 at 04:13):

The "official" direction is the geometric direction, which is the direction of the left adjoint. The category Set is the category of sheaves on a point. So, a "point" of a topos $X$ is a geometric morphism where the left adjoint goes like $\mathrm{Set} \to X$.

John Baez (Aug 14 2021 at 04:17):

I tend to like the algebraic direction: for example, a model of a Lawvere theory $X$ is a product-preserving functor $X \to \mathrm{Set}$, and similarly we can think of a point of a topos $X$ as a model of $X$ in the category $\mathrm{Set}$.... but it's the right adjoint part of geometric morphism that's a functor $X \to \mathrm{Set}$.

John Baez (Aug 14 2021 at 04:18):

So, I tend to slip up and use the opposite of the "official" direction when thinking about geometric morphisms.

John Baez (Aug 14 2021 at 04:21):

Anyway, my point (pardon the pun) was that there's no subtlety in defining the objects of the category of pointed toposes: you already know what a pointed topos is.

Mike Shulman (Aug 14 2021 at 05:43):

@Keith Peterson Yes. In general, any (2-)functor $F:C\to D$ induces a functor on coslice categories $c/F : c/C \to F(c)/D$.

David Michael Roberts (Aug 14 2021 at 05:58):

John Baez said:

The "official" direction is the geometric direction, which is the direction of the left adjoint. The category Set is the category of sheaves on a point. So, a "point" of a topos $X$ is a geometric morphism where the left adjoint goes like $\mathrm{Set} \to X$.

The geometric direction is that of the right adjoint...

David Michael Roberts (Aug 14 2021 at 05:59):

And the rest of what you said needs the left and right swapped :-/

Morgan Rogers (he/him) (Aug 14 2021 at 12:12):

Yeah @John Baez sounds like you've got the directions mixed up!

John Baez (Aug 15 2021 at 21:44):

Okay, I can never get left and right straight in this game. I give up. I should get back to working on something where the difference between left and right doesn't matter.