stub for *Blakers-Massey theorem*. Need to add more references…

I have given the definition of *(infinity,1)-pretopos* an entry, from appendix A of Jacob Lurie’s “Spectral Algebraic Geometry”.

Since this is defined via a variant of the Giraud-Rezk-Lurie axioms with *some* of the infinitary structure made finitary, and since there is a large supply of non-Grothendieck examples (the sub-$\infty$-categories of coherent objects in any Grothendieck $\infty$-topos are $\infty$-pretoposes), it would be interesting if $\infty$-pretoposes were examples of elementary (infinity,1)-toposes, for some definition of the latter.

Are they?

]]>Since it touches on several of the threads that we happen to have here, hopefully I may be excused for making this somewhat selfish post here.

For various reasons I need to finally upload my notes on “differential cohomology in a cohesive ∞-topos” to the arXiv. Soon. Maybe by next week or so.

It’s not fully finalized, clearly, I could spend ages further polishing this – but then it will probably never be fully finalized, as so many other things.

Anyway, in case anyone here might enjoy eyeballing pieces of it (again), I am keeping the latest version here

]]>I noticed that there was no entry *quotient stack*, so I quickly started one, just to be able to point to it from elswhere.

created super infinity-groupoid

(to be distinguished from smooth super infinity-groupoid!)

currently the main achievement of the page is to list lots of literature in support of the claim that the site of superpoints is the correct site to consider here.

]]>created a stub for *twisted differential cohomology* and cross-linked a bit.

This for the moment just to record the existence of

- Ulrich Bunke, Thomas Nikolaus,
*Twisted differential cohomology*(arXiv:1406.3231)

No time right now for more. But later.

]]>have tried to brush-up the entry locally infinity-connected (infinity,1)-topos.

Kicked out a bunch of material that we had meanwhile copied over to their dedicated entries and tried to organize the remaining material a bit better. Need to work on locally infinity-connected site

]]>I was looking for a place to record a somewhat more global overview of the notion of *locally presentable category*, its related notions and its generalizations to higher category theory. But somehow all of the existing entries feel too narrow in focus to accomodate this. So I ended up creating now a new entry titled

Think of this as accomodating material such as one might present in a seminar talk that is meant to bring people with some basic background up to speed with the relevant notions, without going into the wealth of technical lemmas.

I only just started. Will continue in a moment after a short break…

]]>I have split off the section on *points-to-pieces transform* from *cohesive topos* and expanded slightly, pointing also to *comparison map between algebraic and topological K-theory*

Am working on the entry *higher Cartan geometry*. Started writing a *Motivation* section.

This is just the first go, need to quit now, will polish tomorrow.

]]>Given an $\infty$-topos, an object $X$, and a $f$ 1-monomorphism (i.e. (-1)-truncated) is the internal hom $[X,f]$ again a 1-mono?

And dually for $f$ 1-epimorphism (i.e. effective epimorphism) do we have some extra conditions such that $[f,X]$ is 1-mono?

]]>I have added to *Postnikov tower* paragraphs on the *relative* version, (*definition* and *construction* in simplicial sets).

I also added the remark that the relative Postnikov tower is the tower given by the (n-connected, n-truncated) factorization system as $n$ varies, hence is the tower of *n-images* of a map in $\infty Grpd$. And linked back from these entries.

I came to think that the pattern of interrelations of notions in the context of locally presentable categories deserves to be drawn out explicitly. So I started:

Currently it contains the following table, to be further fine-tuned. Comments are welcome.

| | | inclusion of left exaxt localizations | generated under colimits from small objects | | localization of free cocompletion | | generated under filtered colimits from small objects |
|–|–|–|–|–|—-|–|–|
| **(0,1)-category theory** | (0,1)-toposes | $\hookrightarrow$ | algebraic lattices | $\simeq$ Porst’s theorem | subobject lattices in accessible reflective subcategories of presheaf categories | | |
| **category theory** | toposes | $\hookrightarrow$ | locally presentable categories | $\simeq$ Adámek-Rosický’s theorem | accessible reflective subcategories of presheaf categories |
$\hookrightarrow$ | accessible categories |
| **model category theory** | model toposes | $\hookrightarrow$ | combinatorial model categories | $\simeq$ Dugger’s theorem | left Bousfield localization of global model structures on simplicial presheaves | | |
| **(∞,1)-topos theory** | (∞,1)-toposes |$\hookrightarrow$ | locally presentable (∞,1)-categories | $\simeq$ <br/> Simpson’s theorem | accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories |
$\hookrightarrow$ |accessible (∞,1)-categories |

Given a differential cohesive topos $\mathbf{H}$, then for each object $X \in \mathbf{H}$ there is the jet comonad $Jet_X \colon \mathbf{H}_{/X} \to \mathbf{H}_{/X}$, which is a right adjoint (to the infinitesimal disk bundle operator). Therefore by this proposition its Eilenberg-Moore category of coalgebras $EM(Jet_X)$ is itself a topos, over $\mathbf{H}_{/X}$.

**Question 1:** is the topos $EM(Jet_X)$ cohesive, if $\mathbf{H}$ is?

Actually, that seems unlikely due to the dependency on $X$. Reminds one of the tangent topos construction. Therefore:

**Question 2:** Do the toposes $EM(Jet_X)$ as $X$ ranges over $\mathbf{H}$ maybe glue into one big topos? If so, is *that* cohesive if $\mathbf{H}$ is?

This question is motivated from discussion of variational calculus here.

(Incidentally, by Marvan 86 we have that $EM(Jet_X) = PDE(X)$ is the category of partial differential equations with variables in $X$.)

]]>I am currently writing an article as follows:

*Classical field theory via higher differential geometry*

AbstractWe discuss here how the refined formulation of classical mechanics/classical field theory (Hamiltonian mechanics, Lagrangian mechanics) that systematically takes all global effects properly into account – such as notably non-perturbative effects, classical anomalies and the definition of and the descent to reduced phase spaces – naturally is a formulation in “higher differential geometry”. This is the context where smooth manifolds are allowed to be generalized first to smooth orbifolds and then further to Lie groupoids, then further to smooth groupoids and smooth moduli stacks and finally to smooth higher groupoids forming a higher topos for higher differential geometry. We introduce and explain this higher differential geometry as we go along. At the same time as we go along, we explain how the classical concepts of classical mechanics all follow naturally from just the abstract theory of “correspondences in higher slice toposes”.This text is meant to serve the triple purpose of being an exposition of classical mechanics for homotopy type theorists, being an exposition of geometric homotopy theory for physicists, and finally to serve as the canonical example for and seamlessley lead over to the formulation of a local prequantum field theory which supports a localized quantization to local quantum field theory in the sense of the cobordism hypothesis.

This started out as a motivational subsection of *Local prequantum field theory (schreiber)* and as the nLab page *prequantized Lagrangian correspondences*, but for various reasons it seems worthwhile to have this as a standalone exposition and as a pdf file.

I am still working on it. Section 1 and two have already most of the content which they are supposed to have, need more polishing, but should be readable. Section 3 is currently just piecemeal, to be ignored for the moment.

]]>Concerning our geologically slow discussion elsewhere, in various other threads, on $\infty$-toposes that contain non-trivial stable homotopy theory.

Here is a trivial thought:

while we grew fond of identifying the $\infty$-category and allegedly $\infty$-topos of parameterized spectra as the tangent (infinity,1)-category to $\infty Grpd$, maybe it’s after all more useful to think of it instead as the full sub-category of the slice $(\infty,2)$-category

$(\infty,1)Cat_{/Spectra}$on the $(\infty,0)$-truncated objects, which we are inclined to write

$\infty Grpd_{/Spectra} \hookrightarrow (\infty,1)Cat_{/Spectra} \,.$But suppose these two obviously plausible facts about $(\infty,2)$-toposes hold true:

slices of $(\infty,2)$-toposes are $(\infty,2)$-toposes;

full subcateories on $(\infty,0)$-truncated objects inside $(\infty,2)$-toposes are $(\infty,1)$-toposes

then it would follow immediately that $\infty Grpd_{/Spectra}$ is an $(\infty,1)$-topos.

Maybe somebody could remind me why this obvious (naive?) strategy for going about it is no good.

]]>The Hinich-Pridham-Lurie theorem on formal moduli problems says that unbounded $L_\infty$-algebras over some field are equivalently the (“infinitesimally cohesive”) infinitesimal $\infty$-group objects over the derived site over that field.

May we say anything about an analogous statement for infinitesimal group objects in the tangent $\infty$-topos of the $\infty$-topos over the site of smooth manifolds?

There exists for instance the $L_\infty$-algebra whose CE-algebra is

$\{ d h_3 = 0, d \omega_{2p+2} = h_3 \wedge \omega_{2p} | p \in \mathbb{Z} \} \,.$This looks like it wants to correspond to the smooth parameterized spectrum whose base smooth stack is the smooth group 2-stack $\mathbf{B}^2 U(1)$, and whose smooth parameterized spectrum is the pullback along $\mathbf{B}^2 U(1) \to \mathbf{B}GL_1(\mathbf{KU})$ of the canonical smooth parameterized spectrum over $\mathbf{B}GL_1(\mathbf{KU})$, for $\mathbf{KU}$ a smooth sheaf of spectra representing multiplicative differential KU-theory.

So it looks like it wants to be this way. How much more can we say?

]]>I added the bare statement of the list of conditions to *Artin-Lurie representability theorem*, and then added the remark highlighting that the clause on “infinitesimal cohesion” implies that the Lie differentiation of any DM $n$-stack at any point is a formal moduli problem, hence equivalently an $L_\infty$-algebra. Made the corresponding remark more explicit also at *cohesive (∞,1)-presheaf on E-∞ rings*.

I gave André Joyal’s lectures in Paris last week their own category:reference page on the $n$Lab, in order to be able to link to them conveniently (from entries such as topos theory and (infinity,1)-topos theory):

]]>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).

am starting *complex analytic infinity-groupoid* (in line with “smooth infinity-groupoid” etc.) and *higher complex analytic geometry*. Currently there is mainly a pointer to Larusson. To be expanded.

I have been writing about synthetic formalization of (super-)Einstein equations of motion in various talk notes scattered around, (such recently at *Modern Physics formalized in Modal Homotopy Type Theory (schreiber)*) but now it is maybe time to turn it into a polished comprehensive account.

It would seem that I should be doing this in a separate document, but since the proof depends on a fairly wide range of chapters in cohesion, I have been typing it now right into the big file. A first version is now in section 7.1.3 “Gravity” of the dcct pdf. This needs polishing, but that’s where it is for the moment.

]]>Given any object $V$ in an elementary $\infty$-topos, then its automorphism group $\mathbf{Aut}(V)$ may be characterized as the pointed connected object which is the 1-image of the name of $V$ in the type universe.

Now if $V$ itself has group structure, hence if there is pointed connected $\ast \to \mathbf{B}V$, what is the elementary way to speak of its group $\mathbf{Aut}_{Grp}(V)$ of group automorphisms, hence of automorphisms of $\mathbf{B}V$ that preserve the basepoint?

And I’d need a canonical forgetful map

$\mathbf{Aut}_{Grp}(V) \longrightarrow \mathbf{Aut}(V) \,.$ ]]>at *Mayer-Vietoris sequence* I had once recorded (previous nForum discussion is here) some observations about the generality in which Mayer-Vietoris-like homotopy fiber sequences exist in infinity-toposes.

In the section *Over an infinity-group* are discussed sufficient conditions such that for $f : X \to B$ and $g: Y \to B$ two morphisms in an $\infty$-topos to an object $B$ that carries $\infty$-group structure, we obtain a long homotopy fiber sequence of the form

The discussion in Example 2 in that section in that entry gives this under the assumption that the $\infty$-topos is 1-localic, hence that it has a 1-site of definition. Or rather, in that case another discussion gives that every group object in the $\infty$-topos is presented by a presheaf of simplicial groups, and the discussion at *Mayer-Vietoris sequence* takes that to deduce the above fiber sequence.

But all this smells like a hack for a statement that should be true more generally. In which generality do we have the above statement?

]]>Given an $\infty$-topos $\mathbf{H}$ with a comonad $\flat$ on it and given a pointed connected object $\mathbf{B}G$, write $\theta_G \colon G \to \flat_{dR} \mathbf{B}G$ for the homotopy fiber of the homotopy fiber of the counit $\flat \mathbf{B}G \to \mathbf{B}G$.

I’d like to characterize the internal automorphism group $\mathbf{Aut}(\theta_G)\in Grp(\mathbf{H})$ of $\theta_G$ regarded as an object in $\mathbf{H}^{\Delta^1}$, hence the group whose global points are diagrams in $\mathbf{H}$ of the form

$\array{ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR} \mathbf{B}G \\ {}^{\mathllap{\simeq}}\downarrow &\swArrow^{\simeq}& \downarrow^{\mathrlap{\simeq}} \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR} \mathbf{B}G }$There will be a map from $\mathbf{Aut}^{\ast/}(\mathbf{B}G) = \mathbf{Aut}_{Grp}(G)$ to this group in question. Is this an equivalence?

I was thinking this should be easy, but now maybe I am being dense.

]]>