Re: Topos objects

[Alexander Gietelink Oldenziel <a.f.d.a.gietelinkoldenziel@gmail.com>]

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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