Oh lord, right.

Oh, also just realized I meant to say "constant functor valued in the terminal object of D" but anyway... that's irrelevant.

Yeah so I guess given C/X there's not really any "diagram" that we can use to describe X? I mean, I guess we could do something like push-forward to Top to get a best approximation?

@JonathanBeardsley Your basic intuition isn’t wrong: one can express the shape of a sheaf on a topos as a colimit, but it is a colimit in the 2-category of toposes, not in the 2-category of categories.

Let E be a topos, and let X be a sheaf on E (i.e. X is an object of E). Then X determines a geometric morphism X : E -> S, where S denotes the “object classifier” topos, whose points are infinity-groupoids.

The shape of X is the colimit of the geometric morphism X : E -> S in the 2-category of toposes. That is, the left Kan extension in this 2-category of the morphism X : E -> S along the unique morphism E -> 1 (where 1 denotes the terminal object of this 2-category).

(Note that the shape of X doesn’t exist (as an infinity-groupoid) in general; it does if the topos E is locally contractible. Otherwise, it is a pro-infinity-groupoid.)

In more familiar terms, all this just says that the “shape functor” from the category (of sheaves on) E to the category of infinity-groupoids is the left adjoint to the “constant sheaf” functor.

(I’m freely swapping between thinking of a topos as a space and as a category (of sheaves on that space), so I hope that made sense.)

@AlexanderCampbell That's pretty neat! Let me see if I understand. I'm guessing "the shape of $X$" is the same as "the shape of $E/X$", i.e. the pro-$\infty$-groupoid given by composing the left and right adjoint of the geometric morphism $E/X \to Top$, right? And you're saying this can be identified with a certain geometric morphism $1 \to S$ (obtained by LKE), i.e. a certain $\infty$-groupoid. So when the shape is not an $\infty$-groupoid, does this LKE not exist?

@TimCampion That’s right. I wonder if there’s a way to talk about pro-infinity-groupoids inside the 2-category of toposes?

Maybe something can be done using the proarrow equipment (geometric morphisms, lex functors)...

@TimCampion The pro-infinity-groupoids are the proarrows from 1 to 1?

Yeah. I'm trying to recall what the appropriate "equipment-y" version of LKE would be...

I'm not sure how much that would buy anyway -- I don't really know what the "geometric" interpretation of that proarrow equipment is.

Neither do I. In an equipment, proarrows from 1 to 1 arise as “homs” of objects... but I don’t see how to think of the shape of the topos in this way.

Maybe one can make it work by using ~*waggles eyebrows*~ condensed mathematics.

Although perhaps in this case the Barwick-Haine Pyknotic mathematics stuff makes more sense.

Ah here's something: ncatlab.org/nlab/show/shape+of+an+%28infinity%2C1%29-topos

Hi guys, excuse me for the 'elementary' question.

Suppose you have a factorization system (S,T) on C. Then I understood from HTT that the category of factorizations (f,g) of a given h is a contractible Kan complex.

What's C here, an ∞-category?

Buut, it should be the case, at least for inert-active factorization system in infinity operads, that given a square diagram in which the left vertical map is in S and the right vertical map is in T, then the space of liftings is a contractible Kan complex, no matter if the horizontal maps are in the factorization system (as one requires for complete factorizations). Whiy this should be the case?

Yeah is it

I am sorry, can you reformulate? Are you asking why the inert-active factorization system is a factorization system?

Because the property you asked about is the definition of left orthogonal in HTT.5.2.8.1

No no! I mean: if (S,T) is a factorization system, then S is left orthogonal to T. It should be the case, moreover, that the space of liftings in a given orthogonality diagram is contractible!

References are HTT 5.2.8.8 (definition), 5.2.8.17 (contractibility of complete factorizations).

That's clearer now?

What do you mean with "left orthogonal"?

Can i make diagrams here in chat?

It's kind of hard, I never quite understood how

I usually create a small image on my computer and upload it

However, f:x \to y is left orthogonal to g: x' \to y if for every \alpha:x \to x', \beta : y \to y' , there exist an h: y \to x' such that it makes everything commute

That's not the definition in HTT

uhm

HTT.5.2.8.1 asks explicitly that the space of lifts is contractible, not just non-empty

ok

This is different from the definition used when you set up model categories. Ofc they are related, but not in a completely straightforward way

then ok, I can read it through HTT, HA

@DenisNardin That's where I got lost

Thanks Denis! :)

Yeah, I imagined it was something like that :)

No problem

However, I have seen that you are in Paris 13! It's where I attended my homotopy classes this year

And before I studied in Pisa :)

Oh you are in Paris? Did you take Yonatan and Gregory's classes?

It's always good to find a fellow Pisano :)

Yes! Exactly

And Vallette's one in the previous semester

They have done a great path to the infinity in my opinion, even if of course the technical details are always a bit harsh to me (as you have seen :) )

Yeah, this stuff is pretty hard, and I don't think we have worked out the correct exposition quite yet

You mean like HTT and HA are a bit "not-that-introductory"?

You can say that... They are well written (mostly, there are a couple of sections in HA I find not so good), but they're very much "from an expert, to the experts"

Which is what they're supposed to be, so we can't complain too much. I realize it comes as a bit of a shock when you start reading the literature on your own for the first time

My first year of phD was a bit nightmarish (so much stuff to learn...)

Well yes, I feel the same. They are well written but certainly not usable for a soft introduction. I am trying to get just the general picture and incresingly better the details. A good thing about HTT, HA is that after a few readings, even if details are not clear, one can somehow grasp what's going on because of introductory paragraphs and precise references.

I think most people can be sympathetic to that :)

Well, thanks again|

!

See ya

See you and good luck!

@skd I have never heard of a book or document of the kind you mention. What makes you believe it exists?

@CharlesRezk i saw it cited in peter may's "Applications and generalizations of the approximation theorem" and in mahowald's "Some homotopy classes generated by ηj". in the former, it is cited as a preprint, but the latter says that it's been "submitted to Springer Lecture Notes Series", and specific sections of that source are even cited (so it must have been fairly complete)

i was able to prove the statements i wanted which were supposed to be in that document, but i'm guessing there are more interesting results in there

I see. I found another probably reference to it, in N. Kuhn, "Geometry of James-Hopf maps". I wonder what the story is about it.

You can try asking a question on the main site. Kuhn is sometimes active there, he might know

ok, will do. thanks @CharlesRezk and @DenisNardin

i posted the question at mathoverflow.net/questions/334095/…

What would a good "soft introduction" to higher category theory look like?

@CharlesRezk Higher categories as in infinity categories, or higher categories in general?

As in infinity categories.

If the former, I would be happy to see examples of how they make some things much easier. For example, I have found that Denis have posted some very helpful answers on Mathoverflow. Ideally it would show how to work with the objects and maybe defer some of the technical details to later. I have read a lot of material about different sorts of fibrations, but I still feel far from being able to work with infinity-categories proficiently.

Ideally, it would contain many simple examples. I think describing taking Postnikov truncation as applying a certain right adjoint enlightening. The equivalence between fibrations and certains infinity-functors. These are not deep examples, but still, kinda neat and show how it makes certain constructions very clean

Ideally, it would probably explain a lot of the basic notions of topology from an infty-categorical perspective. Spectra, for example, and the smash product. Proving that spaces are generated by a point under homotopy colimits

And then it would contain some stuff on why algebraic geometers should care, something on derived categories and stable infinity categories. Some examples close to an algebraic geometer’s heart

I do not know much though, but this is something that would interest me, but speaks to my tastes more as an algebraic geometer...

In my small and modest opinion the problem is that right now before starting with the "fun" stuff there's a huge amount of technical work that needs to be done to set the theory on a sure footing. But going through all that can be a bit of a slog, and obscure the relative simplicity of the basic ideas. Also the proof of straightening-unstraightening is really hard, but until someone finds a better one we are sort of stuck with it, and for an introduction it might be ok to blackbox it

I don't know how to set up a better introduction though...

Oh, I would want to see how tp construct the symmetric monoidal structure on the derived category of a scheme using infinity-cats. I have not seen it, but would like to know how it goes.

I think the "ideal" soft introduction will be audience-dependent -- maybe it's best to think of the goal as being how to best tailor the material to one's audience. So if the audience consists of homotopy theorists, the exposition should look very different from what you'd say to an audience of algebraic geometers.

It might almost be easier to introduce to non-homotopy-theorists in the sense that there are fewer "bad" habits to "unlearn".

Although I suppose if your audience is algebraic geometers, they may be just as used to thinking about derived functors in a model-dependent way as homotopy theorists are.

But maybe introducing the subject to computer scientists (for HoTT) or combinatorialists (for whatever it is they're doing with 2-Segal spaces) would offer more of a clean slate

I guess I'm saying in my ideal world, $\infty$-categorical ideas are introduced in concert with homotopy-theoretic ideas.

Also I want to heavily emphasize the scare quotes I put around "bad" above -- I don't really think model-dependent thinking is "bad"!

Although come to think of it -- this is kind of analogous to the situation with learning ordinary category theory. It's a common complaint among senior undergraduates / junior grad students that one is typically expected to learn category theory through a kind of osmosis. For many people, it seems to work out, but I think there are still many others who never become comfortable with, say, the notion of adjoint functors. It's an expository challenge which maybe still hasn't been fully solved.

@TimCampion This is the wrong room to comment, but whatever. I think the issue with my statement comes from the notion of "$C^0$ approximation". The notion in Gromov's h-principle doesn't produce maps $f, f': U \to J^1 \Bbb R^{n,1}$ that are $C^0$ close, but rather such that $f'(K)$ lies in an already fixed neighborhood of $f(K)$, and $f' = (g,dg)$ a map to the jet space that arises from an actual timelike map to $\Bbb R^{n,1}$.

You imagine that $f'$ is a very wrinkly map.

This confusion has caused trouble for me before so though I haven't pinned down the precise correction to what I said, it's probably this. Sorry for the false end.

@MikeMiller All I know is that you can do some very counterintuitive with the $h$-principle, constructing very wrinkly maps. So I wouldn't rule out that your original conclusion might be correct. (FWIW Mike and I are discussing this question -- the issue which we haven't fully resolved is what the $h$-principle says about timelike curves in Minkowski space)

@TimCampion Well, I thought I remembered causal curves coming up in this discussion before, but I spent a lot of time searching before my comment and never found someone mention the specific case. Plus, your argument seems valid to me.

@NicholasKuhn by the way, Sanath was wondering where he could find a book by Mahowald and Unell which is referenced in your paper on the geometry of the James-Hopf map.

Here's what I really want to know about Lorentz manifolds: given a Lorentz manifold $M$, there should be a sort of "exit path"-type construction on $M$: an $\infty$-category $Exit(M)$ whose points are those of $M$, and where $Map_{Exit(M)}(x,y)$ is the space of piecewise-smooth (say) timelike paths from $x$ to $y$. Only I'm a little worried that this might not capture quite all the structure one would want -- maybe you also want to record the data of lightlike paths... and it becomes a bit murky