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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 10th 2013
• (edited Oct 10th 2013)

have added a paragraph tangent infinity-category – Tangent infinity topos meant to extract the argument from Joyal’s “Notes on Logoi” that the tangent $\infty$-category of an $\infty$-topos is an $\infty$-topos. Then a remark on how this should imply that the tangent $\infty$-topos of a cohesive topos is itself cohesive over the tangent base $\infty$-topos.

I am not making any claims tonight, just sketching an argument. Hope to come back to it tomorrow when I am awake again.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

Is that argument still looking good in the cold light of day?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 11th 2013

It seems so. Thomas Nikolaus says he sees an alternative proof in terms of excisive functors. Will be discussing this a little later today.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

Thomas Nikolaus says he sees an alternative proof in terms of excisive functors.

Is this related to Charles Rezk’s comment on polynomial functors?

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

Since Goodwillie calculus speaks of n-excisive (∞,1)-functors, I started that, linking it to the $n=1$ entry Urs just started excisive (∞,1)-functor.

Note that an addition at Goodwillie calculus on ’∞-Toposes of polynomial functors’ links to polynomial functors where it says

This entry is about a notion in category theory. For a different notion of the same name in (stable) homotopy theory see at Goodwillie calculus.

So I guess we need some disambiguation.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013
• (edited Oct 11th 2013)

Oh, so is there a ’$k$-jet $(\infty, 1)$-category’ as a higher-tangency form of ’tangent $(\infty, 1)$-category’?

Hmm, but you’re bring jets in here.

EDIT: Of course, this has come up already

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 11th 2013
• (edited Oct 11th 2013)

Yes, right, I guess $n$-excisive diagrams in an $\infty$-topos will also form an $\infty$-topos again.

The argument by Joyal is really very general and simple in itself: it just says that every lex reflective sub-type of diagrams in an $\infty$-topos forms an $\infty$-topos (this is clear) and that all those sub-classes which can be obtained by a filtered $\infty$-colimits hve this property (but this is also clear). That argument applies very generally. And whenever the diagram category has a terminal object does the resulting diagram topos look like the codomain fibration with some structure put on each slice.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 11th 2013
• (edited Oct 11th 2013)

Concerning jets: yes, I agree it seems plausible to speak of the “$n$-jet $\infty$-topos” of an $\infty$-topos. But of course these “jets” are only analogous to the jets to an actual geometric space that might be available inside the $\infty$-topos.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

In the same way that “tangent ∞-topos” is only analogous to the tangent to an actual geometric space that might be available inside the ∞-topos? There’s nothing wrong with analogous analogies.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 11th 2013

Right! sure, nothing wrong with it, on the contrary. I just wanted to clarify that where elsehwhere I mention jets inside a cohesive $\infty$-topos, that it’s a different concept, albeit analogous.

Sorry, maybe that was a pointless comment of mine.

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

So cohesion is burgeoning:

• Plain, Euclidean, smooth, infinitesimal
• Super versions of these (except Euclidean?)
• Global cohesion
• Tangents to all the above, and $k$-jets
• Iterated tangents
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 11th 2013
• (edited Oct 11th 2013)

Hi David,

now I finally have a quiet minute. Some leftover replies:

Concerning the super-version of Euclidean topological cohesion: one can certainly define $SuperCartSp_{Top}$ and the $\infty$-topos over it will be cohesive. But it will turn out not to differ in an interesting way from the $\infty$-topos over just $CartSp_{top}$. The reason is that the super-extension is really a graded-infinitesimal extension and there are no “topological infinitesimals”. Technically: a map from the odd line to any other super-space if given by choosing a derivation of the function algebra of that other space, and there are no non-trivial such on algebras of functions on topological spaces. Otherwise there would be vector fields in topology without the presence of a smooth structure.

Concerning the space of examples of cohesion: yes, it is looking good. There are many things one should now say about tangent cohesion. The foremost is maybe this: the intrinsic cohomology of a tangent $\infty$-topos is twisted generalized cohomology, in fact twisted bivariant cohomology. So the differential cohomology in a tangent cohesive $\infty$-topos will automatically by twisted differential cohomology. This used to be something that was open. Uli Bunke and Thomas Nikolaus will have an article out about it soon.

Another thing that Uli Bunke, Thomas Nikolaus and Michael Völkl will have out soon is the formulation of differential cohomology in stable cohesion. They observe that under stabilization, the axioms of cohesion have even better consequences. Notably the following fact holds true in stable cohesion (hence in the subcategory of a tangent cohesive $\infty$-topos of spectrum objects paraneterized over the points):

for every cohesive spectrum $A$, the $\Pi$-modality naturality square of the cofiber of the $\flat$-modality counit is a homotopy pullback, i.e.

$\array{ A &\longrightarrow& A/\flat A \\ \downarrow &(pb)& \downarrow \\ \Pi(A) &\longrightarrow& \Pi(A/ \flat A) } \,.$

This just follows from the axioms of cohesion in the stable context. And this is a great fact. On the one had it is great because it gives a precise charaxcterization of how every cohesive stable homotopy type is a twisted product of its underlying stable homotopy type $\Pi(X)$ with what remains of its geometric piece after removing the flat bit.

On the other hand, this pullback diagram can be (and now is being) interpreted as being the very archetype of why there is a homotopy pullback in the traditional definition of generalized differential cohomology in the first place. One can take the standpoint that it is this diagram that exhibits every cohesive stable homotopy type $X$ as being the differential cohomology refinement of the underlying generalized cohomology theory $\Pi(X)$ by the “differential form” datum $X/\flat X$. In examples they show that this is precisely what comes out.

So in conclusion, it seems we only just opened the door to the world of cohesion.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeOct 11th 2013

I have made a note of that pullback fact here:

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeOct 11th 2013

Gosh, things move quickly!

in fact twisted bivariant cohomology

So does the tangent idea explain why twisted bivariant cohomology appears in motivic quantization?

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeOct 12th 2013

Interesting – looks like another kind of fracture theorem?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeOct 12th 2013

So does the tangent idea explain why twisted bivariant cohomology appears in motivic quantization?

Good point, yes I am rethinking now how some of the discussion of twisted bivariant geometric generalized cohomology theory simplifies in $T \mathbf{H}$. It seems to come out very nicely, but I need to think about it just a bit more.

@Mike: oh, right. That’s a great point. Hm…

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeOct 13th 2013

I am feeling that there must be a slick way to express the smash product closed monoidal structure on

$Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$

in terms of the Cartesian monoidal structure on $T \mathbf{H}$ and some base change. But right now I don’t fully see it yet. Anyone an idea?

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeOct 13th 2013
• (edited Oct 13th 2013)

To warm up ,let

$X \in \mathbf{H} \hookrightarrow T \mathbf{H}$

be a homotopy type and

$E \in Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$

be a spectrum object. Then what is the internal hom

$[X,E] \in T \mathbf{H}$

?

I suppose it is in fact the correct mapping spectrum, by $\infty$-Yoneda.

• CommentRowNumber19.
• CommentAuthorMike Shulman
• CommentTimeOct 14th 2013

Well, you can get the smash product on $T_* \mathbf{H}$ from the external-smash-product monoidal structure on $T\mathbf{H}$ using base change (that’s in May-Sigurdsson and was a major motivation for the theory of monoidal fibrations), but the latter is different from the cartesian monoidal structure (although it agrees with it after projection down to $\mathbf{H}$). You probably know all that, but it makes your question seem a bit unlikely to me.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeOct 14th 2013

Thanks, Mike. I was looking at May-Sigurdsson, but I have not studied it page-wise and the way it has the emphasis on the models rather than on the general abstract statements, I feel I may still be missing the crucial bits.

More important for me than the smashing is the mapping spectra. Do you agree with me about the following:

• for $X \in \mathbf{H} \hookrightarrow T\mathbf{H}$ and $E \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}$ the Cartesian internal hom $[X,E]$ is the mapping spectrum out of $\Sigma^\infty X$ into $E$?

• more generally for $X$ as before, for $E \in E_\infty(\mathbf{H})$ with $(\widehat {Pic(E)} \to Pic(E)) \in T \mathbf{H}$ its universal line $\infty$-bundle, then $[X, \widehat{Pic(E)}]$ is the spectrum bundle whose base space is the space of $E$-twists $[X, Pic(E)]$ and whose fibers are the $\chi$-twisted $E$-cohomology spectra for all possible twists $E$

?

And is some statement like this maybe in May-Sigurdsson?

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeOct 14th 2013

Mike,

what might be a way to cite Joyal’s IAS talk (or any other talk) which suggested the approach to stable homotopy type theory via parameterized spectrum objects?

• CommentRowNumber22.
• CommentAuthorMike Shulman
• CommentTimeOct 14th 2013

I don’t think May-Sigurdsson discusses the cartesian internal hom at all.

I don’t remember the title of Joyal’s talk, but you could look through the list of recorded talks on the IAS web site (it might have been recorded), or you could ask him.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeOct 14th 2013
• (edited Oct 14th 2013)

Thanks for the info.

Yes, I looked through through the IAS website, but all I find is a line “Synthetic stable homotopy theory” under “open problems” here.

But, sure, I’ll contact André myself. Let me not waste your time. Thanks again.

• CommentRowNumber24.
• CommentAuthorMike Shulman
• CommentTimeOct 14th 2013

Ok, I spent lunchtime today puzzling over the cartesian internal-homs of parametrized spectra and I think I figured them out. Let $E$ and $F$ be spectra parametrized over base spaces $A$ and $B$, respectively. Internally I would write these as $E:A\to Spectra$ and $F:B\to Spectra$. Then:

• The base space of the cartesian internal-hom is the space of maps $E\to F$ of parametrized spectra. That is, the space whose points consist of a map $f:A\to B$ together with a map $\phi:E\to F$ lying over $f$. Internally this is $\prod_{a:A} SpMap(E_a, F_{f(a)})$.

• The fiber spectrum of the cartesian internal-hom over $(f,\phi)$ is obtained by pulling $F$ back to $A$ along $f$, then taking the spectrum of sections. Internally this is $\prod_{a:A} F_{f(a)}$ where the $\prod$ denotes the product of spectra. Note that this does not depend on $\phi$!

In particular:

• If $E_a = 0$ for all $a$, and $B=1$ (your first example), then the base space is $1$, and the fiber is $\prod_{a:A} F$, which as you said is $SpMap(\Sigma^\infty_+ A, F)$.

• If $A=B=1$ (so both objects lie in the “stable part”), then the base space is $SpMap(E,F)$, and the fiber spectrum over every point is $F$.

• If $E_a = 0$ for all $a$, and $F_b=0$ for all $b$ (so both objects lie in the “unstable part”), then the base space is $Map(A,B)$ and all fibers are again $0$. Thus, the inclusion of the unstable part is a cartesian closed functor.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeOct 14th 2013
• (edited Oct 14th 2013)

Thanks, Mike, that’s excellent!

In particular your second item also reproduces my second item from #20. So thanks for the sanity check!

This second item should noteworthy: that twisted cohomology forms a graded ring spectrum which is graded over the Picard $\infty$-groupoid of twists is an important fact when one wants to use twisted cohomology functorially, such as when defining differential twisted cohomology. It’s a big deal in some corners and usually regarded as somewhat subtle, it seems. Here in $T \mathbf{H}$ it just comes out automatically. I think this is a very nice fact about the tangent $\infty$-topos.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeOct 15th 2013
• (edited Oct 15th 2013)

I have briefly put your last comment into the entry here, in the slick type-theoretic notation.

This needs much more polishing etc., but I need to call it quits now. More tomorrow.

• CommentRowNumber27.
• CommentAuthorMike Shulman
• CommentTimeOct 15th 2013

Thanks; I’ve added a proof in type-theoretic language.

I would suggest trying not to mix the external and internal notations. For instance, I would not write $E:Spectra(\mathbf{H})$. Externally, $E$ is a spectrum object in $\mathbf{H}$, but internally to $\mathbf{H}$ (which is what you mean when writing a typing judgment) it is just “a spectrum”, so the thing to write is $E:Spectrum$.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeOct 15th 2013

Thanks! That’s very nice to see in detail.

I have added some hyperlinks to the proof text. At one point I simply made type-theoretic axiom of choice a redirect to axiom of choice. But eventually it would be nice to have a separate entry on this, I think. (We may have talked about this before.)

• CommentRowNumber29.
• CommentAuthorDavid_Corfield
• CommentTimeOct 15th 2013

Can you see how Goodwillie’s remark works?

There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are flat and torsion-free. There is a map between them, so it is meaningful to subtract them.

Say, we are working in parameterized spectra as $T(\infty Grpd)$. Is the idea that given $E:A\to Spectra$ and a morphism $f: A \to B$, the two tangent connections will transport $E$ to two maps $B \to Spectra$ which can then be subtracted? Would one connection involve the direct image?

• CommentRowNumber30.
• CommentAuthorDavid_Corfield
• CommentTimeOct 16th 2013

More ramblings on a theme:

The linearization and other approximations involved in the calculus of functors are described in several places as a kind of sheafification. E.g., for the manifold calculus, de Brito and Weiss write

we recast the Taylor tower as a tower of homotopy sheafifications.

Is this related to the fact that the tangent $(\infty, 1)$-category and k-jet $(\infty, 1)$-categories (#6 above) of an $(\infty, 1)$-topos are also $(\infty, 1)$-toposes?

Could cohesion also play a role? One of the cases of the calculus of functors is the orthogonal calculus, which involves those orthogonal spectra we heard about recently.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeOct 16th 2013
• (edited Oct 16th 2013)

Hey David,

these are all excellent questions I think, and I agree that generally it must be true that there is a good story relating cohesion to Goodwillie calculus which is to be told here. Unfortunately right now my knowledge of the latter is too journalistic for me to be able to say anything substantial. Maybe somebody else here could say more, that would be nice. Myself, I will need to find some spare time and learn a bit before I can say more along these lines.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeOct 16th 2013
• (edited Oct 16th 2013)

$\array{ && \Pi_{dR} \Sigma^{-1} A && \longrightarrow && \flat_{dR}\Sigma A \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \flat (\Pi_{dR} \Sigma^{-1}A )&& && A && && \Pi(\flat_{dR}\Sigma A) \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && \flat A && \longrightarrow && \Pi A } \,,$

with brief comments. (That this is implied by stable cohesion is the central insight of of the article by Uli Bunke and Thomas Nikolaus to appear.) Will expand on the entry more later, when I have more time.

Notice how the maps of Lawvere’s “extra axioms” appear at the top and the bottom…

• CommentRowNumber33.
• CommentAuthorDavid_Corfield
• CommentTimeOct 16th 2013
• (edited Oct 17th 2013)

Re #31, what’s less than journalistic knowledge?

Anyway, I guess I’m hoping for something like a functor $F: C \to D$ between toposes being $k$-excisive if it factors through $J_k(C)$, the $k$-jet topos, and a polynomial approximation $P_k(F)$ arising via Kan extension of $F$ along $C \to J_k(C)$.

Hmm, maybe I should mention $J_k(D)$ too.

• CommentRowNumber34.
• CommentAuthorDavid_Corfield
• CommentTimeOct 17th 2013

If

the intrinsic cohomology of a tangent $\infty$-topos is twisted generalized cohomology, in fact twisted bivariant cohomology,

what is the the intrinsic cohomology of a $k$-jet $\infty$-topos?

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeOct 17th 2013
• (edited Oct 17th 2013)

Hi David. Good question. While I think nobody has looked at this at all, on general grounds one can say at least the following. (I am a bit rushed, but if the following seems at all close to readable, maybe you’d enjoy pasting it into the Idea-section of the nLab entry.)

So here is the general idea:

given an $\infty$-topos $\mathbf{H}$, then also its arrow $\infty$-category $\mathbf{H}^I$ is an $\infty$-topos, over $\infty Grpd^I$ and it also sits over $\mathbf{H}$ by the codomain fibration, constituting an “extension” of $\mathbf{H}$ by itself:

$\array{ \mathbf{H} \\ \downarrow^{\mathrlap{incl}} \\ \mathbf{H}^I \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,.$

The intrinsic cohomology of $\mathbf{H}^I$ under this fibration is nonabelian twisted cohomology as discussed in some detail in Principal ∞-bundles – theory, presentations and applications (schreiber).

Notice that “stable cohomology”, which is traditionally called generalized (Eilenberg-Steenrod) cohomology may be thought of as the lowest order Goodwillie approximation to nonabelian cohomology: where a cocycle in nonabelian cohomology is a map to any homotopy type, a cocycle in generalized (Eilenberg-Steenrod) cohomology is a map into a stable homotopy type.

In this sense the tangent (infinity,1)-topos $T \mathbf{H}$ is the lowest order linear approximation to the codomain fibration

$\array{ \Stab(\mathbf{H}) \\ \downarrow^{\mathrlap{incl}} \\ T\mathbf{H} \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,.$

From this it is clear that whatever we may say in detail about the $k$th-jet $\infty$-topos $J^k \mathbf{H}$, its intrinsic cohomology is a version of twisted cohomology which is in between nonabelian cohomology and stable i.e. generalized (Eilenberg-Steenrod) cohomology.

It seems that a layered analysis of nonabelian cohomology this way in higher homotopy theory should eventually be rather important, even if it hasn’t received any attention at all yet (as far as I am aware). It seems plausible that a generalization of Chern-Weil theory which approximates classes of principal infinity-bundles not just by universal characteristic classes in ordinary cohomology and hence in stable cohomology, but that one wants to consider the whole Goodwillie Taylor tower of approximations to it.

• CommentRowNumber36.
• CommentAuthorDavid_Corfield
• CommentTimeOct 17th 2013

maybe you’d enjoy pasting it into the Idea-section of the nLab entry.

Yes, but which entry? Should we start a new one for $k$th-jet $(infinity, 1)$-topos?

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeOct 17th 2013
• (edited Oct 17th 2013)

For the momemt one might just put it into tangent (infinity,1)-topos itself (as an outlook putting things in perspective) or into twisted cohomology.

• CommentRowNumber38.
• CommentAuthorDavid_Corfield
• CommentTimeOct 17th 2013

I’m trying to get a better sense of what’s there already.

We have a separate entry tangent cohesive (∞,1)-topos. Then “tangent (infinity,1)-topos” redirects to “tangent (infinity,1)-category”, which has a subsection “Tangent ∞-topos of an ∞-topos” and a sub-subsection “Cohesive tangent ∞-topos of a cohesive ∞-topos” and then later as an example, a sub-subsection “Of an ∞-topos”.

I think putting #35 there would add to the confusion.

So I added it at twisted cohomology, though I don’t know how well it fits. Does the twisted cohomology corresponding to tangent toposes capture all there is about twisted cohomology? If not, it probably needs more explanation.

• CommentRowNumber39.
• CommentAuthorMike Shulman
• CommentTimeOct 17th 2013

I got lost somewhere; do we actually have a definition of “$k$-jet $(\infty,1)$-topos”?

• CommentRowNumber40.
• CommentAuthorDavid_Corfield
• CommentTimeOct 17th 2013

Hmm, not exactly. The closest we got was at #7. I kind of imagined by analogy to the tangent case, something like the fibrewise stabilisation of the fibration of $n$-cubes in $C$ onto evaluation at the terminal object of the cube.

• CommentRowNumber41.
• CommentAuthorDavid_Corfield
• CommentTimeOct 17th 2013

Modulo getting the definition right :), might there be a left adjoint to the fibration at the initial corner, as with the cotangent complex?

• CommentRowNumber42.
• CommentAuthorDavid_Corfield
• CommentTimeOct 17th 2013

A $k$-jet bundle is a subbundle of the k-iterated tangent bundle, isn’t it? I wonder if this carries over somehow. There are no worries about the iterated tangent category construction, which preserves topos-ness and cohesiveness.

• CommentRowNumber43.
• CommentAuthorUrs
• CommentTimeOct 17th 2013

Mike, the idea is to phrase Joyal’s observation in terms of 1-excisive functors and then observe that it goes through for $k$-excisive functors, too.

• CommentRowNumber44.
• CommentAuthorDavid_Corfield
• CommentTimeOct 18th 2013
• (edited Oct 18th 2013)

I was thinking there should be a multivariate calculus of functors, and of course there is, see sec. 6.2. So we have a notion of n-excisive for n = $(n_1, ..., n_m)$.

But presumably there should be a multivariate jet construction. Thinking in SDG terms, we need Kock’s $D_k(n) = [(x_1, ..., x_n) \in R^n|$ product of any $k+1$ is zero].

• CommentRowNumber45.
• CommentAuthorDavid_Corfield
• CommentTimeOct 18th 2013

With all this dash to the $(\infty, 1)$ case, I’d forgotten Urs had already being doing tangents for $(n, n)$-categories way back here and here. Did the extra generality get used, or did it mostly boil down to the $(n, 1)$ case?

• CommentRowNumber46.
• CommentAuthorDavid_Corfield
• CommentTimeOct 20th 2013

Urs #43, I don’t quite see how things go here. In the $n = 1$ case, the idea is that there’s a connection between 1-excisive functors and the tangent $(\infty, 1)$-category construction.

For the latter, I see how that diagram seq gets used to define parameterized spectra, and I see how that delivers the fiberwise stabilization of the codomain fibration, $Stab(Func(\Delta[1],C) \to C)$.

So how does this relate to 1-exciveness, which concerns taking pushout squares to pullback squares? Is the relation via the squares of seq?

If so, when you say we do the same for $k$-excisive functors, is it that seq gets replaced by a higher dimensional with $k+1$-cubes instead of squares? And what is the equivalent of $Stab(Func(\Delta[1],C) \to C)$? As I had it in #40?

• CommentRowNumber47.
• CommentAuthorCharles Rezk
• CommentTimeOct 20th 2013
• (edited Oct 20th 2013)

I understand things like this:

I write $\mathrm{Top}$ for the homotopy theory of spaces (i.e., $\infty$-groupoids). Let $\mathcal{F}=\mathrm{Func}^{\mathrm{filt}}(\mathrm{Top}_*,\mathrm{Top})$ denote the homotopy theory of functors from pointed spaces to spaces which commute with all filtered colimits. We can identify a full subtheory $\mathcal{F}_n\subset \mathcal{F}$ of $n$-excisive functors.

Goodwillie (in “Calculus 3”) has given an explicit formula to construct the left adjoint $P_n\colon \mathcal{F}\to \mathcal{F}_n$ to the inclusion $\mathcal{F}_n\subset \mathcal{F}$, called “$n$-excisive approximation”. It’s immediate from Goodwillie’s formula that $P_n$ commutes with finite homotopy limits, and therefore each $\mathcal{F}_n$ is an $\infty$-topos. Thus we obtain a tower of $\infty$-topoi $\mathcal{F}\to \cdots \to \mathcal{F}_n\to\mathcal{F}_{n-1}\to \cdots\to \mathcal{F}_1\to \mathcal{F}_0$.

$\mathcal{F}_0$ consists of constant functors, and so is equivalent to $\mathrm{Top}$, while $\mathcal{F}_1$ is $1$-excisive functors; Goodwillie showed that $(\mathcal{F}_1)^{P_0=X}$, the $1$-excisive functors $F$ equipped with an identification of $P_0F$ with the space $X$, is the same as parameterized spectra over $X$.

“Intrinsic” characterizations of the $\mathcal{F}_n$ for $n\gt1$ seem to be much harder to get at. Probably one can say that $(\mathcal{F}_n)^{P_0=X}$ is equivalent to “parameterized reduced $n$-excisive functors over $X$”, where “reduced $n$-excisive functors” are $(\mathcal{F}_n)^{P_0=*}$. It is hard to give a characterization of $(\mathcal{F}_n)^{P_0=*}$ for $n\gt1$ that is satisfying, though Arone and Ching now have a decent answer to this, I think, in terms of comonads on spectra.

Correction. When I originally wrote this, I wrote $\mathrm{Func}^{\mathrm{filt}}(\mathrm{Top},\mathrm{Top})$ for the definition of $\mathcal{F}$, a category of functors on unbased spaces, thus perpetuating a standard confusion in this subject. There is perfectly good theory of $n$-excisive approximations for functors on unbased spaces, and it gives rise to a nice tower of $\infty$-toposes (that construction is very general)— but this tower is somewhat different than the one I actually described above. For instance, “unbased $1$-excisive functors” are the same thing as “parameterized spectra $E\to X$ equipped with a section”.

• CommentRowNumber48.
• CommentAuthorUrs
• CommentTimeOct 20th 2013

Charles,

thanks for pushing this further!

This would be theorem 1.8 in “Calculus III”, I suppose.

Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”.

• CommentRowNumber49.
• CommentAuthorCharles Rezk
• CommentTimeOct 20th 2013

Fully explicitly, it is in theorem 7.1.1.10 of Lurie’s “Higher Algebra”.

I suppose I should read that book someday.

• CommentRowNumber50.
• CommentAuthorUrs
• CommentTimeOct 20th 2013

Oh, and in remark 7.1.1. 11 right below Lurie attributes the statement of your #47 above to you. :-)

Is this in print anywhere? Do you say this in “Homotopy toposes”? (I don’t remember having seen it there, but I may have forgotten.)

• CommentRowNumber51.
• CommentAuthorUrs
• CommentTimeOct 20th 2013

Goodwillie showed that […] the 1-excisive functors F equipped with an identification of P0F with the space X, is the same as parameterized spectra over X.

Maybe you can save me a minute: which page is this statement on?

• CommentRowNumber52.
• CommentAuthorCharles Rezk
• CommentTimeOct 20th 2013

Urs 51: I’m not sure. I think it was in Calculus 3, but that would have been a preprint version I saw many years ago.

Urs 50: I told Jacob that, but I’d learned it from Biedermann.

• CommentRowNumber53.
• CommentAuthorCharles Rezk
• CommentTimeOct 20th 2013

Urs 51. I’m probably hallucinating. I don’t know where he would have said this, though it is true.

Goodwillie does briefly discuss parameterized spectra (in Calculus 1, remark 1.6). Here he is thinking about functors $\mathrm{Top}/X\to \mathrm{Top}_*$ (it would be the same I think if you consider functors $X\backslash \mathrm{Top}/X\to \mathrm{Top}$), and in this case reduced $1$-excisive functors are the same as parameterized spectra over $X$.

However, this is a completely different appearence of parameterized spectra, than the one I discussed above. (Basically, an object of $\mathcal{F}_1$ is a parameterized spectrum over the space $T=F(*)$ which is the value of $F$ at the terminal object of the domain; 1-excisive functors on the slice category $X\backslash\mathrm{Top}/X$ want to be parameterized over $X$, which is itself the terminal object of the domain.)

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeOct 20th 2013

I see, thanks for the information. I suppose it is about time that I contact Georg Biedermann.

• CommentRowNumber55.
• CommentAuthorDavid_Corfield
• CommentTimeOct 21st 2013

About those two flat tangent connections mentioned in #29, Goodwillie says in the abstract for a 2005 workshop

their difference is the tensor ﬁeld known as smash product of spectra.

He continues

I will say something about higher-order jets and about differential operators. I cannot make much sense of differential forms (except 0-forms and 1-forms), but I may talk about them anyway.

• CommentRowNumber56.
• CommentAuthorUrs
• CommentTimeOct 21st 2013
• (edited Oct 21st 2013)

Have now finally contacted Georg Biedermann. He kindly points out his articles where he constructs model categories of n-excisive functors, which are homotopy toposes in Charles’s terminology, hence present the $\infty$-toposes that we are discussing.

While the Lab is unresponsive, I’ll record these reference here:

• CommentRowNumber57.
• CommentAuthorUrs
• CommentTimeOct 24th 2013

I have now added some notes on this to section 4.1.2 of https://dl.dropboxusercontent.com/u/12630719/cohesivedocument131024.pdf

• CommentRowNumber58.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

’accomodate’ twice

’projetion’

Since $T \mathbf{H}$ is differentially cohesive over a cohesive $\mathbf{H}$, but $\mathbf{H}^{\Delta[1]}$ is not, at what point in the $J^n(\mathbf{H})$ interpolating between them does this property go?

By the way, it’s not just altruism that has me reading your book. I’ve promised to give a talk with the grand title Homotopy Type Theory: a revolutionary language for philosophy of logic, mathematics, and physics?. I’m nervous about getting the physics part up to scratch. Think I might talk about covariance. Maybe also something on quantization.

• CommentRowNumber59.
• CommentAuthorUrs
• CommentTimeOct 24th 2013
• (edited Oct 24th 2013)

Thanks!

Concerning “accomodate”, isng’t that correct? I checked with Google and Google seems to agree? Did you maybe make a reverse typo when reporting a typo?

Concerning $\mathbf{H}^I$: this is in fact cohesive over $\mathbf{H}$, but my section on it doesn’t mention that (yet). Back when I wrote that section I used $\mathbf{H}^I$ for other purposes than giving twisted cohomology an internal home. It is now only through the company of the flashy $T \mathbf{H}$ that its superficially boring cousin $\mathbf{H}^I$ is getting some recognition, too.

Concerning your talk: interesting! If you have any preliminary notes, I’d be happy to look through them and return a tiny bit of the favor.

BTW, that same week I will be speaking in Edinburgh on “Higher toposes of laws of motion”.

• CommentRowNumber60.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

Google must be picking up similar misspellings then. It’s double c and double m: accommodate. (Latin: ad-commodus, probably in turn con-modus)

Yes $\mathbf{H}^I$ is cohesive, but is it differentially cohesive?

That would be great if you would look over especially the physics slides (when they’re ready).

• CommentRowNumber61.
• CommentAuthorUrs
• CommentTimeOct 24th 2013

Oh, I see. Yes, my fault. Fixed now.

Concerning differentially cohesive: ah, now I see what you mean. No, not over $\mathbf{H}$, unless I am missing something. True.

What makes one kind of cohesive homtopy types $X$ be “differential cohesive” over another $x$ is if they are “infinitesimal thickenings” of that latter, meaning that you cannot see the thickening of $X$ when homming into it out of $x$; something non-trivial only happens the other way around, homming $X$ into $X$ sees “tangents” in $x$.

• CommentRowNumber62.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

Yes I knew that it wasn’t differentially cohesive as the modalities don’t equate. But the original question was then where in the tower of cohesive jet toposes does differential cohesiveness fail. If $J^1 = T$ is differential, does it already fail at $J^2$?

• CommentRowNumber63.
• CommentAuthorUrs
• CommentTimeOct 24th 2013
• (edited Oct 24th 2013)

I need to think about those higher jet toposes more in order to answer, but meanwhile just one comment to clear up terminoloy (since I may have cause a mixup here)

I am saying

Now if $\mathbf{H}$ is infinitesimally cohesive over $\infty\mathrm{Grpd}$, then the inclusion $\infty Grpd \hookrightarrow \mathbf{H}$ also exhibits $\mathbf{H}$ as being differentially cohesive over $\infty Grpd$. But the converse does not in general hold.

• CommentRowNumber64.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

I guess I was hoping that exponentiation by $I$ is simple enough that the failure of $\infty Grpd^I$ to be infinitesimally cohesive would imply that $H^I$ is not differentially cohesive for cohesive $H$.

• CommentRowNumber65.
• CommentAuthorUrs
• CommentTimeOct 24th 2013

Yes, indeed it’s not. Sorry, I thought I already said this. In a way it is the finite length of the interval $(0 \to 1)$ that prevents it, for it forces the neighbouring adjoints to be different, which for $T \mathbf{H}$ coincide.

By the way, I have now added on p. 256 more remarks previewing the way that the homotopy cofiber of $\infty$-toposes along a differential cohesion inclusion produces the underlying infinitesimal cohesion, here https://dl.dropboxusercontent.com/u/12630719/cohesivedocument131024.pdf.

This is quite probably true in full generality, but right now I just show it in the examples.

• CommentRowNumber66.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2013

Now I’m not sure what the ’it’ of

indeed it’s not

refers to. Or the ’this’ of

I was expecting $\mathbf{H}^I$ not to be an infinitesimal thickening of $\mathbf{H}$. I was wondering if this fact could already be seen from the failure of $\infty Grpd^I$ to be infinitesimally cohesive over $\infty Grpd$.
It’s part of my transporting thickenings idea. Were $\mathbf{H}^I$ a thickening over $\mathbf{H}$ we could transport it to $\infty Grpd^I$ over $\infty Grpd$. But we know we can’t.