CAUTION: The Sender of this email is not from within Dalhousie. 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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