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following Mike’s suggestion, I have split off from cohesive topos the entry
I used that splitting-off to play Bourbaki and decide that I don’t follow Lawvere’s definition in all detail. Instead, it seems to me we can usefully streamline it. It should say just two things: a cohesive $(\infty,1)$-topos is
locally and globally $\infty$-connected
local .
And that’s it. That gives the quadruple of adjoint functors, where the inverse image and its parallel functor are both full and faithful.
I have also added an Interpretation-section where I highlight that this implies two central properties of cohesive $(\infty,1)$-toposes:
they have the shape of the point in the sense of shape of an (infinity,1)-topos (this is implied by local plus global $\infty$-connectedness);
they look like small neighbourhoods of the standard point (this is what the locality axioms means, given the standard examples for local toposes).
Mostly as a reminder to myself, I state the following question:
I feel that to possibly fine-tune the definition of cohesive $(\infty,1)$-topos, it would be helpful to better understand to which extent classical example of toposes that are like “thick points” fail to be cohesive (if they do). The $\infty$-versions of the standard examples of “small” local toposes in the literature: under which conditions are they locally $\infty$-connected?
If most of them do happen to be locally $\infty$-connected, then it should follow that the crucial axiom that makes a point-like $\infty$-topos qualify as “big” is another one. Maybe this is all in the axiom that $p_* \to p_!$ is a mono. But I do not understand yet what geometric condition that axiom really models.
Presumably there are comparison maps between cohesive $(\infty, 1)$-toposes ($(\infty, 1)$-cohesive geometric morphisms?) to compare these thickened points.
Are there many other flavours of thickened point? Examples so far seem to favour the reals. Can you have holomorphically thickened points or p-adic or adelic ones? How about points with symmetry [*/G]?
Right, there must be many more interesting examples. The ones I have so far all look similar only because I find them by using the same proof strategy with which I prove $(\infty,1)Sh(CartSp)$ to be cohesive.
One immediate thing that fails for sites that look like actual points, such as $*//G$ is that they fail to satisfy the condition that there are finite products in the site. Without this my proof does not show that $\Pi$ preserves products. But what is worse about $*//G$ is that it does not even have a terminal object: apart from the Grothendieck topology being suitable, the existence of a terminal object in the site is most important: this is what makes everything local and gives $Codisc$.
What I personall would like to understand are these examples:
let $T$ be any abelian Lawvere theory. When can I find a sensible Grothendieck topology on $T$ that makes it cohesive?
For proving this for $T = CartSp$ I am using quite a bit of technology for good open covers, as we have discussed here at length. These statements hardly have direct analogs for general $T$. But something else could do. This is because we know that generally the objects of $T$ look like products $\mathbb{A}^k$ of the canonical $T$-line object $\mathbb{A}^1$. So intuitively they should all be contractible.
Eventually I need to know this in derived geometry: what are sites of simplicial $T$-algebras such that the $\infty$-sheaves over it still form a cohesive $\infty$-topos?
For the kind of application that I am following, this is a pressing question. In particular I need to understand what the functor $\Pi$ does exactly on such cohesive derived $\infty$-toposes. But currently I do not have much of a handle on this question.
Re: morphisms, we need to be a bit careful with the terminology, since according to the usual naming scheme for toposes and geometric morphisms, a “cohesive geometric morphism” $E\to S$ would be one which makes $E$ into a cohesive $S$-topos – which is different from if you have two cohesive $S$-toposes $E\to S$ and $F\to S$ and you want a morphism $E\to F$ over $S$ which “preserves cohesion.” I guess the natural thing to require of a geometric morphism that “preserves cohesion” is that it commute with the extra adjoints $\Pi$ and $Codisc$ in some way.
My initial reaction to the idea of p-adic cohesion is that it wouldn’t work because unlike the reals, the p-adics are not locally contractible (or even locally connected).
My initial reaction to the idea of p-adic cohesion is that it wouldn’t work because unlike the reals, the p-adics are not locally contractible (or even locally connected).
Ok. I’m just wondering what we are forced to accept if we agree to ’Physics takes place in a cohesive $\infty$-topos’. I’m always intrigued to see how the reals pop up seemingly out of nowhere. We had a chat about that in the post on locally compact abelian groups. John says there
It’s a bit more magical the way $\mathbb{R}^n$ stands out in the classification of locally compact Hausdorff abelian groups. If I’m not mistaken, the reason is that the $\mathbb{R}^n$’s are the only groups of this sort that are connected and lack compact subgroups.
I’m wondering what rabbits have been put into the hat of cohesive $\infty$-toposes.
I think this is a quite interesting aspect that David is raising there. Currently I am lacking a sufficient survey of classes of examples of cohesive $\infty$-toposes to make much progress on answering this question, but the general question is very interesting:
once we have – walking in Lawvere’s footsteps – convinced ourselves that we have found a general abstract theory that is close to having the necessary and sufficient properties to serve as a context for structures of fundamental physics, we can ask which models this theory has. And which properties these models have. Maybe we find that all these models share an aspect of physics that does not explicitly appear in the definition of the theory.
So I’d really enjoy following up on these questions. But right now I am lacking a good strategy for how to attack this problem further. But I’ll keep it in mind and maybe eventually we see something.
Currently I am lacking a sufficient survey of classes of examples of cohesive $\infty$-toposes to make much progress on answering this question
Math Overflow question?
Sure, I’ll post one.
okay, after polishing cohesive (infinity,1)-topos a bit more (expanded the beginning of the introction, added two more subsections to the Properties-subsection, restructured the Examples-section) I posted to MO:
I was about to post a reply to Mike’s latest comment on the Cafe, but that seems to be down now. Lest I lose my message, I post it here for the moment:
Mike wrote:
Now how do Lie algebras come into the picture?
The discussion I gave should still hold, only that $Lie \mathbf{B}G$ is bigger than I said it was.
So for every $L_\infty$-algebra $\mathfrak{g}$ there is an object $\mathbf{B}\mathfrak{g}$ in the $(\infty,1)$-Cahiers topos. It is modeled by the simplicial presheaf
$\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \mathfrak{g}_k$where $\mathfrak{g}_k$ is an infinitesimal object (with a single point) and $\mathbf{\Delta}$ is the fat simplex .
How do we recognize this intrinsically as an $\infty$-Lie algebra? Like this:
it is a fat point, $\Gamma \mathbf{B}G \simeq * \simeq \Pi \mathbf{B}G$;
it has the $\mathbb{A}^1$-cohomology of the $L_\infty$-algebra $\mathfrak{g}$ (where $\mathbb{A}^1$ is our line object, which is the real line for the Cahier topos):
$\pi_0\mathbf{H}(\mathbf{B}\mathfrak{g}, \mathbf{B}^n \mathbb{A}^1) \simeq H^n(CE(\mathfrak{g}))$
(this follows with the technology described at function algebras on $\infty$-stacks).
So the question now is what is the universal factorization of morphisms $\mathbf{B}\mathfrak{g} \to \mathbf{B}H$? If we assume we do have the coreflector that we have been talking about, then that will produces it, $\mathbf{B}\mathfrak{g} \to Lie \mathbf{B}H \to \mathbf{B}H$.
And given the above simplicial presheaf model of $\mathbf{B}\mathfrak{g}$ and observing that this is cofibrant, we see in a more precise way what we just said two comments earlier: this $Lie \mathbf{B}H$-thing has to consist of the points in $\mathbf{B}H$ and their infinitesimal neighbourhoods, because that’s the only “cohesvive pieces” that the infinitesimal objects $\mathfrak{g}_k$ can hit in each degree.
So first I said we should have $Lie \mathbf{B}H$ is $\mathbf{B}\mathfrak{h}$, but as you remarked rightly, this can’t be. It must be somthing like “$\mathbf{B}(\mathfrak{g} \times \Gamma(G))$”. Not sure yet.
Okay, sounds reasonable; I’ll wait until you figure it out. (-: It does sound like a neat thing, if it is true.
I have polished and slighly expanded the section on infinitesimal cohesion at cohesive (infinity,1)-topos
added also intrinsic discussion of the tangent Lie algebroid in an $(\infty,1)$-topos with infinitesimal cohesion
Urs, is the infinity-category of simplicial objects in a cohesive topos itself a cohesive $(infinity,1)$-topos ?
Hi Zoran,
a sufficient condition for this to be true is that the topos has a site of definition that is not just a cohesive site but even an infinity-cohesive site (by the proof given there).
More precisely, in that case the simplicial localization of the category of simplicial sheaves at the local weak equivalences is a cohesive $\infty$-topos, yes.
I have added the observation (in the section on Cohesive oo-groups), that every $\infty$-group object $G$ in a cohesive topos is presented by a presheaf of simplicial groups, and that every delooping $\mathbf{B}G$ is presented by a simplicial presheaf of the form $\bar W G$.
I have added in the last “Structures”-section Higher holonomy and Chern-Simons functional a general abstract definitions of
of the $\infty$-Chern-Simons theory induced by any characteristic class in the cohesive $\infty$-topos.
There is still a slight gap in the story: I know how to fully abstractly get the $\infty$-Chern-Simons action functional as a morphism in discrete $\infty$-groupoids. However, in order for computing the derived covariant phase space as the homotopy fiber of the differential refinement of the action functional, we need it as an internal morphism on cohesive $\infty$-groupoids. I know how that is obtained for the special case of smooth cohesion, but I do not yet understand the fully general abstract condition under which the external action functional refines to an internal one. So for the moment it just says “In suitable situations this external construction refines to an internal one as follows.”
added to the list of canonical structures in a cohesive $(\infty,1)$-topos its intrinsic variational calculus
so far there’s just the definition and its relation to a buch of keywords. I’ll see if one can say something general abstract about the variational bicomplex from tis point of view
have made explicit a proposition that was used implicitly before: that in a cohesive $\infty$-topos $\mathbf{H}$ we have an equivalence
$\Omega : Connected(\mathbf{H}) \leftrightarrow Grp(\mathbf{H}) : \mathbf{B}$in the section Cohesive oo-groups
Pointed connected or just connected?
Right, pointed connected, of course. I have fixed it.
added at cohesive (infinity,1)-topos in the section Infinitesimal paths the abstract definition of formally smooth objects : those objects $X$ for which $X \to \mathbf{\Pi}_{inf}(X)$ is an effective epimorphism. I point out that
this is the evident generalization of the condition as considered by Simpson-Teleman
and that by just slightly rewriting it it becomes equivalent to the abstract condition that Kontsevich-Rosenberg consider in the context of Q-categories (see the section on formal smoothness there).
Using Kontsevich-Rosenberg we can then also say in every cohesive $\infty$-topos with infinitesimal cohesion:
$X$ is formally unramified if $X \to \mathbf{\Pi}_{inf}(X)$ is some notion of monomorphism;
$X$ is formally étale if $X \to \mathbf{\Pi}_{inf}(X)$ is an equivalence.
Wait a second with the last paragraph. Formally etale is stronger than formally unramified.
Sorry, I switched the two conditions: formally unramified is the mono-condition. Formally etale is the equivalence conditions. So formally etale = equivalence = mono + epi = formally smooth + formally unramified.
(I’ll change this typo in my comment above.)
Right. I have introduced a somewhat related entry adjoint triple. It would be good to have more properties listed for various kinds of adjoint triples eventually.
I have introduced a somewhat related entry adjoint triple.
Okay, good. I have added some links to and from it. I have also created a stub for adjoint quadruple. There is a section at cohesive topos with generalities on adjoint triples and quadruples, that should be moved to the new entries, then.
I have added more to adjoint triple and created Frobenius functor. One should clarify more to other Frobenius notions.
Right, it will be good eventually have some properties of adjoint triples in general, which do not require cohesive topos setup separated.
Wikipedia Frobenius algebra entry explains a relation between Frobenius algebras and Frobenius extensions. The induction functor for the latter generalized historically to Frobenius functors…
For info: There is a newly published paper by McCurdy and Street in the latest Cahiers (vol LI-1) on ‘What separable Frobenius monoidal functors preserve.’
Tim, Frobenius monoidal functors are NOT instances of Frobenius functors. F. monoidal f. is sort of symmetric in way in having both monoidal and comonoidal (lax and colax in different terminology) structure. F. f. is about left and right adjoint existing and being mutually equal or isomorphic.
I had read the first fact that you mentioned in the entry, so my point was just to announce that the paper was now published.
OK, I have now cited the arXiv version of McCurdy, Street at Frobenius monoidal functor. If you have the exact/full publication reference and link please add them there.
Done. :-) Thanks.
I have added a subsection cohesive oo-topos – twisted cohomology. Nothing there that is not in principle also at twisted cohomology, but maybe being more stream-lined. It starts out with the over-category-definition that Domenico emphasized is the most intrinsic one, and then unwinds this to the definition in terms of homotopy fibers.
This just for completeness, since further below differential cohomology is defined as a special case of twisted cohomology
I have added a subsection cohesive oo-topos – twisted cohomology.
Have added now also discussion of the equivalence to the definition by way of homotopy classes of sections.
Maybe the server has a bad day, worse than the other days. I had to once more split the entry into pieces, for it to be saved at all. For some reason just splitting it into further two wouldn’t help, so now there is a total of four pieces. Which means that now essentially all the external links to its subsections are broken.
added to the section on differential cohomology more details on the two exact sequences that differential cohomology sits in (curvature exact sequence and characteristic class exact sequence)
(the server is still saving the entry, for several minutes already. If it does not display yet when you read this, try again in a few more minutes).
Just a thought: maybe instead of adding on to a single page more and more until it gets unmanageable and has to be split (breaking subsection links), we should create new entries for new subjects as we go along. In particular, maybe the desire to link to a subsection should alert us that perhaps it is time to separate out a new page.
maybe instead of adding on to a single page more and more until it gets unmanageable and has to be split (breaking subsection links), we should create new entries for new subjects as we go along.
Sure, if that makes sense. Here I felt the page (as a text file) was still quite manageable and that breaking it apart would be unnatural. One can see this also from the fact that there is no way of breaking it apart into stand-alone subentries. One needs to implement by hand some infrastructure for a collection of entries that all need to be read as parts of a whole. This is lots of extra work. I wish I could decide for myself when I want to do this extra work and when not.
I have added definition of extension of cohesive $\infty$-groups and statement and proof of how extended $\infty$-bundles are characterized by $\infty$-bundles on the total space of the bundle that they extend. At the end of the section on cohomology.
For instance, it seems to me more natural, and more in the wiki style (and also more readable, because more organized), for the discussion of infinitesimal cohesion to be on a separate page, given that infinitesimal cohesion is extra structure on a cohesive topos rather than an intrinsic part of the notion.
Agreed.
I have split at cohesive (infinity,1)-topos the Definition-section into three subsections:
internal definition formulated in HoTT.
In 2. I have included the relevant statement that the internal formulation is indeed equivalent to the external one. In 3. I have added some commentary and the codicsrete reflective subcategory.
Missing in the internal description is still the condition that $\Gamma = \tilde \Gamma$. But I have to quit now and continue tomorrow.
This is also discussed on the $n$Café here.
added clarification to the Definition-section at cohesive (infinity,1)-topos as mentioned over on the $n$Café here.
added more details on the internal formulation following Mike’s comment
at cohesive (infinity,1)-topos I have started a new section Properties – Over arbitrary bases with some simple observations on bases of cohesion, to go along with the $n$Café-discussion here.
added a quick paragraph cohesive oo-topos – Examples – Simplicial objects, mostly for the moment as a reminder to myself to come back to this and expand
cohesion at IHES: video recording
A related question on MathOverflow: https://mathoverflow.net/questions/444806/cohesive-structure-of-cahiers-and-dubuc-topoi.
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