am working on a new chapter of the *geometry of physics* cluster:

Still a bit rough towards the end, but I need a break now.

In the course of this I started splitting off a further chapter *geometry of physics – prequantum geometry*, but that’s rudimentary at the moment.

It should be the case that $CRing^{op}$ is coreflective in $(CRing^{\Delta^{op}})^{op}$, the coreflection being 0-truncation of simplical rings

$CRing^{op} \stackrel{\hookrightarrow}{\underset{\tau_0}{\longleftarrow}} (CRing^{\Delta^{op}})^\op \,.$That’s the kind of structure on sites that induces differential cohesion. Need to check some details tomorrow when I am more awake.

This must have surfaced before. For instance a derived analogue of a de Rham stack construction that does not remove nilpotent ring elements, but removes higher simplicial cells in the ring. Has this been discussed anywhere?

(Thanks to Mathieu Anel for discussion today.)

]]>created *anti-reduced object*, for completeness

At some point I had made up the extra axiom/terminology saying that an object $\mathbb{A}^1$ in a cohesive $\infty$-topos “exhibits the cohesion” if the shape modality is equivalent to $\mathbb{A}^1$-localization. Now I was talking about that assumption with Mike and noticed that this didn’t have a reflection on the $n$Lab yet.

So now I have added, for the record, the definition here at “cohesive oo-topos” and cross-linked with the existing discussion at “continuum”.

]]>Sometimes I think that we have too much meta-discussion here. Nervetheless I feel the need to start the following one, apologies in advance. Let’s try to not turn it into an all too lengthy discussion.

Over in another thread here Andrew and myself both expressed dissatisfaction with the term *smooth space* as I myself have been using it in various entries. I think Andrew and me dislike the word for rather different reasons, but nevertheless it’s suboptimal. I was just reminded of that when editing *variational calculus*.

What would be a better term? I started to think I should instead say

consistently. How do you all feel about that?

]]>We had discussed here at some length the formalization of formally etale morphisms in a differential cohesive (infinity,1)-topos. But there is an immediate slight reformulation which I never made explicit before, but which is interesting to make explicit:

namely I used to characterize formal étaleness in terms of the canonical morphism $\phi : i_! \to i_*$ between the components of the geometric morphism $i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ that defines the differential cohesion – because that formulation made close contact to the way Kontsevich and Rosenberg formulate formal étaleness.

But there is a more suggestive/transparent but equivalent (in fact more general, since it works in all of $\mathbf{H}_{th}$ not just in $\mathbf{H}$) formulation in terms of the $\mathbf{\Pi}_{inf}$-modality, the “fundamental infinitesimal path $\infty$-groupoid” operator:

a morphism $f : X \to Y$ in $\mathbf{H}_{th}$ is formally étale precisely if the canonical diagram

$\array{ X &\to & \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(Y) }$is an $\infty$-pullback.

(It’s immediate that this is equivalent to the previous definition, using that $i_!$ is fully faithful, by definition.)

This is nice, because it makes the relation to general abstract Galois theory manifest: if we just replace in the above the infinitesimal modality $\mathbf{\Pi}_{inf}$ with the finite path $\infty$-groupoid modality $\mathbf{\Pi}$, then the above pullback characterizes the “$\mathbf{\Pi}$-closed morphisms” which precisely constitute the total space projections of locally constant $\infty$-stacks over $Y$. Here we now characterize *general* $\infty$-stacks over $Y$.

And for instance in direct analogy with the corresponding proof for the $\mathbf{\Pi}$-modality, one finds for the $\mathbf{\Pi}_{inf}$-modality that, for instance, we have an orthogonal factorization system

$(\mathbf{\Pi}_{inf}-equivalences\;,\; formally\;etale\;morphisms) \,.$I’ll spell out more on this at *Differential cohesion – Structures* a little later (that’s why this here is under “latest changes”), for the moment more details are in section 3.7.3 of *differential cohomology in a cohesive topos (schreiber)*.

I have started something in an entry

which has grown out of the the desires expressed in the thread *The Wiki history of the universe*.

This is tentative. I should have maybe created this instead on my personal web. I hope we can discuss this a bit. If it leads nowhere and/or if the feeling is that it is awkward for one reason or other, I promise to remove it again. But let’s give this a chance. I feel this is finally beginning to converge to something.

]]>Maybe one day it would be fun to find the time to create as $n$Lab entries all those linked to here:

The WIki History of the Universe,

maybe at least up to the entry “star formation”, which still fits under the “Physics” headline of the $n$Lab (after that geology and then biology kicks in, which we should probably leave to other wikis).

One interesting thought to explore might be if we can expand that link list to the left. ;-)

]]>