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Thanks! You deserve a special medal for chasing this error.
Has anyone noticed that the lifting properties of a certain cohesion setup are similar to the lifting properties for a model category? The “domain and codomain fibration” section at Q-category makes me wonder how model categories might be defined in the langauge of differential cohesion (or just cohesion).
I don’t know what you mean.
Maybe this is a better place to ask the question here. I want to understand the composites of the modalities $\flat,\sharp,\Im,\&$ on a differential cohesive topos. I’ve worked out the following composites so far just from the adjunctions $\flat\dashv\sharp$ and $\Im\dashv\&$, plus $\flat \Rightarrow \&$, where the entry in the table is “$row \circ column$”.
$\begin{array}{ccccc} \downarrow \circ \rightarrow & \flat & \sharp & \Im & \& \\ \flat & \flat & \flat & & \flat \\ \sharp & \sharp & \sharp & & \\ \Im & \flat & & \Im & \&\\ \& & \flat & & \Im & \& \\ \end{array}$I think I’ve convinced myself now that $\sharp\circ\& = \sharp$, because:
$\sharp\circ\& = \nabla_{th} \circ \Gamma_{th} \circ i_* \circ i^! = \nabla_{th} \circ \Gamma \circ i^! \circ \nabla_{th} \circ \Gamma_{th} = \sharp.$This uses $\Gamma_{th} \circ i_* = \Gamma$ and $\Gamma\circ i^! = \Gamma_{th}$, which are Proposition 2.5 on this page differential cohesive (infinity,1)-topos. Does that seem right?
I could compute all the rest except one if I knew that $\Gamma \circ i^* = \Gamma_{th}$. This seems plausible to me because $\Gamma$ and $\Gamma_{th}$ are evaluation of (pre)sheaves at the terminal object and $i^*$ is precomposition with an inclusion, so it ought to hold if $i:C \hookrightarrow C_{th}$ preserves the terminal object. This condition isn’t included in the definition of “infinitesimal neighborhood site” but it seems plausible – does it hold in examples?
If $\Gamma \circ i^* = \Gamma_{th}$ holds then with similar calculations I think I can fill the table out to
$\begin{array}{ccccc} \downarrow \circ \rightarrow & \flat & \sharp & \Im & \& \\ \flat & \flat & \flat & \flat & \flat \\ \sharp & \sharp & \sharp & \sharp & \sharp \\ \Im & \flat & & \Im & \&\\ \& & \flat & \sharp & \Im & \& \\ \end{array}$So the only one that would be missing is $\Im \circ \sharp$. That seems likely to be the most fraught, e.g. in ordinary cohesion $ʃ \circ \sharp$ is strange, kind of like propositional truncation. But whatever it is, it’s an idempotent monad and we can compute its composites with all the other modalities, so if we include it as a primitive we’d have a complete $5\times 5$ table.
$\begin{array}{ccccc} \downarrow \circ \rightarrow & \flat & \sharp & \Im & \& & \Im\sharp \\ \flat & \flat & \flat & \flat & \flat & \flat \\ \sharp & \sharp & \sharp & \sharp & \sharp & \sharp \\ \Im & \flat & \Im\sharp & \Im & \& & \Im\sharp \\ \& & \flat & \sharp & \Im & \& & \Im\sharp \\ \Im\sharp & \Im\sharp & \Im\sharp & \Im\sharp & \Im\sharp & \Im\sharp \\ \end{array}$The composition rules are even easy to remember. When composing $\mu\circ\nu$, if $\mu$ is $\flat$, $\sharp$, or $\Im\sharp$, then the result is $\mu$. Otherwise, the result is $\nu$, unless the composite is $\Im\circ \sharp$ in which case the result is $\Im\sharp$.
What would be a better name for $\Im\sharp$?
Oh hmm, I just found a comment back here that $\Re$ is actually left exact in the intended models? So maybe I should be including it too.
It’s great that you are looking into this!
I’d need to think about the questions you ask. I’ll see if I get to it once some other urgent tasks are out of the way.
But in Def. 3.5.6 of dcct (p. 264) the functor $i \,\colon\, C \hookrightarrow C_{th}$ is required to preserve finite products, hence the terminal object.
This is certainly the case in the standard models of Cartesian spaces included into (graded-)infinitesimally thickened Cartesian spaces: $\mathbb{R}^0$ is the terminal object in all of these sites.
Nice, thanks! That condition seems to be missing from the definition here, should we add it?
Yes, I don’t remember any reason for not including that clause.
I guess $\Gamma \circ i^* = \Gamma_{th}$ would also follow for the same reasons as $\Gamma \circ i^! = \Gamma_{th}$ and $\Gamma_{th} \circ i_* = \Gamma$ if we knew that $i_!$ was left exact, since then $i_! \dashv i^*$ would also be a geometric morphism. And I guess $i_!$ being left exact is equivalent to $\Re$ being left exact, which maybe does hold in the models?
It seems clear that $i \,\colon\, C \hookrightarrow C_{th}$ preserves finite limits in the intended examples/models. This should be sufficient for $i_!$ to be left exact, by the Proposition recorded here. (?)
I think the reduction $\Re$ is not left exact (if I understand the setup correctly, reduction is an endofunctor on the differential cohesive topos). A counterexample is the intersection of a circle with a tangent, which can be computed as a pullback of reduced spaces with a non-reduced result.
Yes, reduction is an endofunctor of the differential cohesive topos, which is $\Re = i_! i^*$ in terms of the adjoint quadruple $i_! \dashv i^* \dashv i_* \dashv i^!$ relating the differential cohesive topos to an ordinary cohesive topos. If what you say is right, Felix, then that means that $i$ doesn’t preserve finite limits, so that $i_!$ isn’t left exact? And so I guess at the level of sites, you would be saying that reduced $C^\infty$-algebras may not be closed under pushout?
I assume this must also be true in pure algebra. For instance, consider the parabola $y=x^2$ and the line $y=0$ sitting inside the affine plane. The corresponding cospan is $R[x,y]/(y-x^2) \leftarrow R[x,y] \to R[x]$, which consists of reduced algebras, but whose pushout $R[x]/(x^2)$ is not reduced. That seems right to me.
(Are we saying “differential cohesive” or “elastic” these days?)
True, my first “clear” statement in #13 was clearly wrong.
I’d like to say “elastic” instead of “differential cohesive”.
Re #17: Something to be aware of is a recent article by Christian Blohmann https://arxiv.org/abs/2301.02583, which uses the adjective “elastic” with a different meaning, but essentially in the same context (sheaves on cartesian spaces).
Yes, when I first saw the title of Blohmann’s preprint, I thought it was adopting the terminology from here, then quickly found it wasn’t.
For what it’s worth, the term “elastic $\infty$-topos” appears on the arXiv in 2020 (arXiv:2008.01101, p. 45)
I have been using it since at least 2017, when it was meant to be the terminology adopted in version 2 of dcct (p. 325 in the version from August 2017).
In August 2018 I gave the lecture series Categories and Toposes (schreiber) at Nesin Math Village, on the progression of cohesive, elastic and solid toposes, for which I used the lecture notes geometry of physics – categories and toposes in which the term “elastic toposes” is used since June 2018.
I understand that a proper publication would be good; the above arXiv:2008.01101 is still waiting for its first referee report, though…
But on absolute grounds, you’ll find that the terminology is beautifully evocative:
“cohesion” for (differential) topology, where points stick together, but loosely,
“elasticity” for differential geometry, where in manifolds the points stick together more stiffly, and where $G$-structure (metric structure) models genuine elasticity phenomena such as in “hearing the shape of a drum”
“solidity” for super-geometry where… but never mind this for the moment.
It’s also a profound hat tip to Lawvere’s original choice of the Hegelian “cohesion”; since after “cohesion” the next quality which Hegel assigns to the “unity of space and time”, as it were, is “elasticity”, saying that this is the place of “Zeno’s dialectic of motion” (as referenced at elasticity – In natural philosophy) — and of course it’s exactly the differential calculus made available in elastic toposes which resolves Zeno’s paradox of motion.
Looking now at Blohmann’s more recent arXiv:2301.02583 it seems he means to understand “elasticity” quite in this spirit. On p. 2 he wants elasticity to provide jet bundle calculus on diffeological spaces. That of course is what is provided by embedding diffeological spaces into the ambient elastic topos, where the jet comonad is the base change comonad along the infinitesimal shape modality. Maybe he might want to cite arXiv:1701.06238 on this topic.
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