Topos objects

Topos objects

[David Roberts]

Dear all,

We know from work of Burroni, Lambek, Macdonald–Stone and Dubuc–Kelley
that toposes are monadic over Graph and over Cat (the 1-category), and
even (and I don't know to whom this is due) 2-monadic over Cat (the
2-category).

I was wondering recently if there is a sensible notion of a *topos
object* in a 2- or bicategory K. One would need suitable structure on
K to support this, I presume finite limits (of the appropriate
weakness) at minimum. I would imagine possibly also an involution akin
to (-)^op, for the following reason. In the work of the above authors,
the definition of topos is taken to be that of a cartesian closed
category with subobject classifier. However, one could take the
terminal object + pullbacks + power objects definition instead,
provided one gave these as functors satisfying certain conditions. And
here I'm not sure if one would take the covariant or contravariant
power object functor. If the latter, we clearly need that involution.

Alternatively, one could take the approach that the relation \in
appearing in the definition of power object is a universal relation,
and one could potentially think of Rel(E) as a suitable completion in
Cat of the putative topos E, and abstract this. It seems a fair bit
more overhead though, and probably more complicated than it's worth
(modulo the fact this whole exercise might also be so!)

My motivation for this, such as it is, is that if one could define
topos objects in a suitable bicategory K, one could take, for
instance, K to be fibrations over a suitable base, or categories and
anafunctors or a combination of these. Maybe it's a sledgehammer
approach, but it seems curious and maybe interesting in its own right.

Best regards,

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Topos objects

[Steve Vickers]

Dear David,

My first reaction when I read your post was that there was an obvious answer  of Yes. But the more I reread it, the more I wondered whether I truly understood what the question was. What did you mean by “topos”? And what would a “topos object” be?

Here’s my initial obvious answer.

If “topos” means elementary topos (optionally with nno), then the theory of toposes is cartesian (aka essentially algebraic), so if “topos object” means “internal topos” (just as eg “group object” means “internal group”), then there is a standard notion of topos object, model of that cartesian theory, in  any 1-category with finite limits. At least, if you take the topos structure to be canonically given.

It seemed too easy. Perhaps you meant something deeper.

First I started to wonder why you stressed the 2-categories. Was it to enable some up-to-isomorphism laxity?

Next I wondered if I was interpreting “topos object” correctly. After all, the theory of toposes has two sorts, for objects and arrows, and an internal topos is carried by two objects. Were you instead thinking of  a single object, with its category structure implied by the ambient 2-cells? For instance, in a 2-category with finite products, a “finite product object” X could be one for which the diagonals X -> 1 and X -> X  x X have right adjoints.

Finally, there is the question of what a “topos” is. If it is akin to a Grothendieck topos, a category of sheaves for a generalized space, then then the relevant structure is that not of elementary toposes, but of “all” colimits and finite limits as in Giraud’s theorem. I find it a very interesting question when an object in a 2-category might have topos structure in that sense.

For example, suppose your 2-category C is that of Grothendieck toposes and geometric morphisms (maps). Then the object classifier O surely is a “topos object” in C, and so also would be O^X for any exponentiable topos X. The points of O^X are the maps X -> O, ie the objects of X, so it is  reasonable to imagine that the topos structure of X might be reflected in the C-internal structure on O^X.

Finitary products and coproducts for O, given by maps O^n -> O, are easy. Eg  for binary products, n=2, map (U, V) |-> UxV. (U, V are points of O, ie sets. The construction is geometric and so does give a geometric morphism.) The infinitary colimits are harder. I conjecture that they are given by M-algebra structure for O, where M is the symmetric topos monad on C. That should  all lift to O^X.

The question with C = Grothendieck toposes generalizes to Grothendieck toposes over S (bounded S-toposes) and - a particular interest of my own - a related 2-category based on arithmetic universes.

All the best,

Steve.

> On 1 Sep 2020, at 03:23, droberts.65537@gmail.com wrote:
> 
> Dear all,
> 
> We know from work of Burroni, Lambek, Macdonald–Stone and Dubuc–Kelley
> that toposes are monadic over Graph and over Cat (the 1-category), and
> even (and I don't know to whom this is due) 2-monadic over Cat (the
> 2-category).
> 
> I was wondering recently if there is a sensible notion of a *topos
> object* in a 2- or bicategory K. One would need suitable structure on
> K to support this, I presume finite limits (of the appropriate
> weakness) at minimum. I would imagine possibly also an involution akin
> to (-)^op, for the following reason. In the work of the above authors,
> the definition of topos is taken to be that of a cartesian closed
> category with subobject classifier. However, one could take the
> terminal object + pullbacks + power objects definition instead,
> provided one gave these as functors satisfying certain conditions. And
> here I'm not sure if one would take the covariant or contravariant
> power object functor. If the latter, we clearly need that involution.
> 
> Alternatively, one could take the approach that the relation \in
> appearing in the definition of power object is a universal relation,
> and one could potentially think of Rel(E) as a suitable completion in
> Cat of the putative topos E, and abstract this. It seems a fair bit
> more overhead though, and probably more complicated than it's worth
> (modulo the fact this whole exercise might also be so!)
> 
> My motivation for this, such as it is, is that if one could define
> topos objects in a suitable bicategory K, one could take, for
> instance, K to be fibrations over a suitable base, or categories and
> anafunctors or a combination of these. Maybe it's a sledgehammer
> approach, but it seems curious and maybe interesting in its own right.
> 
> Best regards,
> 
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts
> Blog: https://thehighergeometer.wordpress.com
> 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Topos objects

[David Roberts]

Dear Steve,

thank you for the thoughtful reply. You are correct in that I did not
mean an internal topos, and I should have specified that at the time
of writing I had in mind an elementary topos.

> Were you instead thinking of a single object, with its category structure
> implied by the ambient 2-cells? For instance, in a 2-category with finite
> products, a “finite product object” X could be one for which the diagonals
> X -> 1 and X -> X x X have right adjoints.

Yes, this is the concept I had in mind. It is less clear to me if, in
addition to finite limits, it would be easier to ask for cartesian
closedness+subobject classifier or for power objects. The latter seems
to me to be simpler, at least in the case when the 2-category admits a
construction analogous to E |---> Rel(E) (not that I know how to do
it, but it seems a reasonable first line of attack). There should be
some 1-arrow in the/an ambient 2-category with an adjoint giving the
power object and the universal relation \in.

One result that would indicate this is on the right track is that a
topos object in Cat should just be an elementary topos (with specified
finite limits, power objects), a topos object in Cat_ana (anafunctors
instead of functors) should be a topos (with *unspecified* finite
limits and power objects), a topos object in Fib(B) should be a fibred
topos, and a topos object in Cat_cocomp should be a cocomplete topos.
I'd be interested to hear (even if off-list) your ideas relating to
arithmetic universes (AUs). I could imagine asking for a topos object
in categories over an AU. Or perhaps in some kind of syntactic
2-category. For an even more exotic example, imagine taking the
2-category of be of the sort Riehl and Verity use for their synthetic
\infty-cateogry work. Then one might argue that a topos object in such
a 2-category is a candidate for an elementary (\infty,1)-topos, since
these 2-categories are meant to have as objects (\infty,1)-categories.

The question of what morphisms to take is then an even more
interesting question. The easiest answer is that one could ask for
logical morphisms. But clearly this is not sufficient for all
purposes. If looking for geometric morphisms I would myself aim for
something like logos morphisms (à la Anel–Joyal), namely left exact
cocontinuous arrows, considering these as morphisms in the opposite
category.

Best regards,

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Wed, 2 Sep 2020 at 20:15, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:
>
> Dear David,
>
> My first reaction when I read your post was that there was an obvious answer of Yes. But the more I reread it, the more I wondered whether I truly understood what the question was. What did you mean by “topos”? And what would a “topos object” be?
>
> Here’s my initial obvious answer.
>
> If “topos” means elementary topos (optionally with nno), then the theory of toposes is cartesian (aka essentially algebraic), so if “topos object” means “internal topos” (just  as eg “group object” means “internal group”), then there is a standard notion of topos object, model of that cartesian theory, in any 1-category with finite limits. At least, if you take the  topos structure to be canonically given.
>
> It seemed too easy. Perhaps you meant something deeper.
>
> First I started to wonder why you stressed the 2-categories. Was it to enable some up-to-isomorphism laxity?
>
> Next I wondered if I was interpreting “topos object” correctly. After all, the theory of toposes has two sorts, for objects and arrows, and an internal topos is carried by two objects. Were you instead thinking of a single object, with its category structure implied by the ambient 2-cells? For instance, in a 2-category with finite products, a “finite product object” X could be one for which the diagonals X -> 1 and X -> X x X have right adjoints.
>
> Finally, there is the question of what a “topos” is. If it is akin to a Grothendieck topos, a category of sheaves for a generalized space, then then the relevant structure is that not of elementary toposes, but of “all” colimits and finite limits as in Giraud’s theorem. I find it a very interesting question when an object in a 2-category might have topos structure in that sense.
>
> For example, suppose your 2-category C is that of Grothendieck toposes and geometric morphisms (maps). Then the object classifier O surely is a “topos object” in C, and so also would be O^X for any exponentiable topos X. The points of O^X are the maps X -> O, ie the objects of X, so it is reasonable to imagine that the topos structure of X might be reflected in the C-internal structure on O^X.
>
> Finitary products and coproducts for O, given by maps O^n -> O, are easy.  Eg for binary products, n=2, map (U, V) |-> UxV. (U, V are points of O, ie sets. The construction is geometric and so does give a geometric morphism.) The infinitary colimits are harder. I conjecture that they are given by  M-algebra structure for O, where M is the symmetric topos monad on C. That  should all lift to O^X.
>
> The question with C = Grothendieck toposes generalizes to Grothendieck toposes over S (bounded S-toposes) and - a particular interest of my own - a related 2-category based on arithmetic universes.
>
> All the best,
>
> Steve.
>
>> On 1 Sep 2020, at 03:23, droberts.65537@gmail.com wrote:
>>
>> Dear all,
>>
>> We know from work of Burroni, Lambek, Macdonald–Stone and Dubuc–Kelley
>> that toposes are monadic over Graph and over Cat (the 1-category), and
>> even (and I don't know to whom this is due) 2-monadic over Cat (the
>> 2-category).
>>
>> I was wondering recently if there is a sensible notion of a *topos
>> object* in a 2- or bicategory K. One would need suitable structure on
>> K to support this, I presume finite limits (of the appropriate
>> weakness) at minimum. I would imagine possibly also an involution akin
>> to (-)^op, for the following reason. In the work of the above authors,
>> the definition of topos is taken to be that of a cartesian closed
>> category with subobject classifier. However, one could take the
>> terminal object + pullbacks + power objects definition instead,
>> provided one gave these as functors satisfying certain conditions. And
>> here I'm not sure if one would take the covariant or contravariant
>> power object functor. If the latter, we clearly need that involution.
>>
>> Alternatively, one could take the approach that the relation \in
>> appearing in the definition of power object is a universal relation,
>> and one could potentially think of Rel(E) as a suitable completion in
>> Cat of the putative topos E, and abstract this. It seems a fair bit
>> more overhead though, and probably more complicated than it's worth
>> (modulo the fact this whole exercise might also be so!)
>>
>> My motivation for this, such as it is, is that if one could define
>> topos objects in a suitable bicategory K, one could take, for
>> instance, K to be fibrations over a suitable base, or categories and
>> anafunctors or a combination of these. Maybe it's a sledgehammer
>> approach, but it seems curious and maybe interesting in its own right.
>>
>> Best regards,
>>
>> David Roberts
>> Webpage: https://ncatlab.org/nlab/show/David+Roberts
>> Blog: https://thehighergeometer.wordpress.com

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Topos objects

[Alexander Gietelink Oldenziel]

Dear Steve,

I believe David is indeed talking about elementary topoi internal to a
suitable bicategory.

Your remarks about the object classifier, exponentiable topoi [image:
S[\mathbb{O}], S[\mathbb{O}]^X] in [image: GTopos/S] are well-taken. I
believe it is sometimes preferred to say they are ' logos'  objects inside
the category of topoi, i.e. formal duals to topoi rather than a topos
object.

For simplicity, let us consider just [image: S[\mathbb{O}]]. It has finite
limits in the sense that for any finite diagram [image: J] we have a
geometric morphism [image: \varprojlim: S[\mathbb{O}]^J \to S[\mathbb{O}]]
which is right adjoint to the diagonal [image: \Delta: S[\mathbb{O}] \to
S[\mathbb{O}]]. Indeed, we may define this simply on points by [image:
\{F(j)\}_{j\in J} \mapsto \varprojlim F(j)], noting that geometric
morphisms preserve finite limits. Similarly, for infinite colimits we have
a morphism [image: \varinjlim: S[\mathbb{O}]^J \to S[\mathbb{O}]], defined
similarly.
To say [image: S[\mathbb{O}]] is a logos object, we should probably say it
satisfies some version of the Giraud axioms. I am not completely confident
how this would work precisely.

I do understand the locale case. Here the Sierpinski locale [image:
\mathbb{S}] is an internal frame in [image: Locale] which simply means we
have [image: \wedge: \mathbb{S}^2 \to \mathbb{S}, \vee_I: \mathbb{S}^I \to
\mathbb{S}] which distribute. The Giraud axioms can also be seen as a sort
of distributivity axioms, but the details elude me.

You mention something about using the M-algebra structure of [image:
S[\mathbb{O}]] where [image: M] is the symmetric topos construction. I am
not sure how the symmetric topos monad will help, could you say more about
this?

best,
Alexander



On Fri, 4 Sep 2020 at 02:54, Steve Vickers <s.j.vickers@cs.bham.ac.uk>
wrote:

> Dear David,
>
> My first reaction when I read your post was that there was an obvious
> answer  of Yes. But the more I reread it, the more I wondered whether I
> truly understood what the question was. What did you mean by “topos”? And
> what would a “topos object” be?
>
> Here’s my initial obvious answer.
>
> If “topos” means elementary topos (optionally with nno), then the theory
> of toposes is cartesian (aka essentially algebraic), so if “topos  object”
> means “internal topos” (just as eg “group object” means “internal group”),
> then there is a standard notion of topos object, model of that cartesian
> theory, in  any 1-category with finite limits. At least, if you take the
> topos structure to be canonically given.
>
> It seemed too easy. Perhaps you meant something deeper.
>
> First I started to wonder why you stressed the 2-categories. Was it to
> enable some up-to-isomorphism laxity?
>
> Next I wondered if I was interpreting “topos object” correctly. After all,
> the theory of toposes has two sorts, for objects and arrows, and an
> internal topos is carried by two objects. Were you instead thinking of  a
> single object, with its category structure implied by the ambient 2-cells?
> For instance, in a 2-category with finite products, a “finite product
> object” X could be one for which the diagonals X -> 1 and X -> X   x X have
> right adjoints.
>
> Finally, there is the question of what a “topos” is. If it is akin to a
> Grothendieck topos, a category of sheaves for a generalized space, then
> then the relevant structure is that not of elementary toposes, but of “all”
> colimits and finite limits as in Giraud’s theorem. I find it a very
> interesting question when an object in a 2-category might have topos
> structure in that sense.
>
> For example, suppose your 2-category C is that of Grothendieck toposes and
> geometric morphisms (maps). Then the object classifier O surely is a “topos
> object” in C, and so also would be O^X for any exponentiable topos X. The
> points of O^X are the maps X -> O, ie the objects of X, so it is
> reasonable to imagine that the topos structure of X might be reflected in
> the C-internal structure on O^X.
>
> Finitary products and coproducts for O, given by maps O^n -> O, are easy.
> Eg  for binary products, n=2, map (U, V) |-> UxV. (U, V are points of O, ie
> sets. The construction is geometric and so does give a geometric morphism.)
> The infinitary colimits are harder. I conjecture that they are given by
> M-algebra structure for O, where M is the symmetric topos monad on C. That
> should  all lift to O^X.
>
> The question with C = Grothendieck toposes generalizes to Grothendieck
> toposes over S (bounded S-toposes) and - a particular interest of my own -
> a related 2-category based on arithmetic universes.
>
> All the best,
>
> Steve.
>

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