Re: Topos objects

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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
> 

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