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I have added to universal covering space a discussion of the “fiber of $X\to \Pi_1(X)$” definition in terms of little toposes rather than big ones.
I find this definition of the universal cover extremely appealing. It seems that this sort of thing must have been on the tip of Grothendieck’s tongue, and likewise of all the other people who have studied fundamental groups and groupoids of a topos, but it all becomes so much clearer (I think) when you state it in the language of higher toposes. In this case, merely (2,1)-toposes are enough, so no one can argue that the categorical technology wasn’t there – so why didn’t people see this way of stating it until recently? Or did they?
Nice.
The intrinsic structure induced by the existence of the $\Pi$-adjoint is just sheer amazing. Elsewhere i started making lists of all the structures that are induced for free from this:
Whitehead towers, Galois theory, van Kampen theory, geometric realization, infinitesimal objects, Lie theory, differential forms, differential cohomology, Chern-Weil theory, Chern-Simons theory,
All this exists for free inside any cohesive $\infty$-topos! And of course to the extent that $(\infty,1)Topos$ looks like one (which we discussed at length on the blog) it holds for that, too.
Or, depending on your point of view, they exist for free for (∞,1)-toposes, and to the extent that objects of a cohesive ∞-topos inherit that structure, they exist for those too. (-:
Okay, sure. I would enjoy that point of view even more if I were more confident that we actually did have all four adjoints available at once on $\infty$-toposes.
For instace take that operation which I said is like Lie differentiation in any $\infty$-connected topos, $\mathfrak{a} := \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} A \to A$, where for a pointed object $A$ the operation $\mathbf{\flat}_{dR}$ is defined by the $\infty$-pullbacl
$\array{ \mathbf{\flat}_{dR} A &\to& Disc \Gamma A \\ \downarrow && \downarrow \\ * &\to& A }$and the operation $\mathbf{\Pi}_{dR} X$ by the $\infty$-pushout
$\array{ X &\to& * \\ \downarrow && \downarrow \\ Disc \Pi X &\to& \mathbf{\Pi}_{dR}(X) } \,.$Using the quadruple of adjunctions in a cohesive $\infty$-topos, one finds notanly that indeed
$\Gamma(\mathfrak{a}) \simeq \Pi(\mathfrak{a}) \simeq * \,.$If all these operations were/are available in $(\infty,1)Topos$ it would give us a remarkably powerful machine to apply to $\infty$-topoess: for instance every $\infty$-topos could be “differentiated” around any of its global points in a sense that has all the relevant formal properties.
So I’d be thrilled if we could do that. But can we do it? It does not seem that we have the quadrupe of adjunctions available on $(\infty,1)Topos$.
We should have it available inside the cohesive $\infty$-topos $(\infty,1)Sh(ContrLocale)$ of $\infty$-sheaves over the site of contractible locales. I’d be inclined to think that that’s a way to embed petit $\infty$-toposes into a cohesive context. But not sure yet to which extent this works out. It is true for ordinary toposes that we can think of them as sitting inside $2Sh(Locales)$. But I don’t know what happens in detail when we restrict to testing with contractible locales. Do you?
I don’t feel any obligation to have such adjoints, or any desire to be able to differentiate an arbitrary topos, since a topos is an incarnation of topology, rather than geometry, and to me differentiation belongs to geometry rather than topology. I would be more inclined to look for a notion of differentiation for certain kinds of structured topos.
However, in the particular pullback and pushout you wrote, you never used the fourth adjoint Codisc, which is the most questionable one for toposes. So I think those diagrams make perfect sense as long as X is locally ∞-connected (or else ΠX means some sort of Galois topos) and A has only a small ∞-groupoid of points, so that a discrete topos on ΓA (the core of the (∞,1)-category of points of A) exists. In fact, I wouldn’t be surprised if the result of that pullback were unchanged under passage to full subgroupoids of ΓA as long as they contain the specified point, in which case it wouldn’t matter whether A has only a small number of points.
However, in the particular pullback and pushout you wrote, you never used the fourth adjoint Codisc,
I do use that: the fact that $\Gamma$ is a left adjoint is used in showing $\Gamma \mathfrak{a} \simeq *$.
But I agree: to the extent that the quadruple of adjunctions is not fully present, it means precisely that there is not quite a notion of geometry, just of topology. This is of course the point of cohesiveness: to axiomatize geometry.
I find it a strangely overspecialized notion of geometry. I mean geometry is a very general subject, much wider than topology, which is just a specialized subsubject. On the other hand, you seem to assume that by geometry in this discussion you call those situations which have most advanced and most symmetric versions of differentiability and more.
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