Homotopy Theory

Saal Hardali

9:06 AM

@DenisNardin Very nice example! Thanks.

Harry Gindi

2:53 PM

@SaalHardali The worst part of it is that some of the stuff in Gaitsgory-Rozenblyum is actually proved

If you take a look at Ayala-Mazel-Gee-Rozenblyum, they give some proofs of some of the stuff in their new appendix.

Also yeah, parametrized spectra are the coolest example of a non-hypercomplete unbounded ∞-topos

because they actually show up in nature

Here's a question though: If I take Shv_fpqc(Aff) and RKE QCoh as QCOH to this category, is QCOH a topos? If not, is there an adjunction between QCOH and Shv_fpqc(Aff)-parametrized spectra?

Aaron Mazel-Gee

4:28 PM

i'll be teaching a first-year grad-level course on algebraic topology this upcoming quarter, and i'm looking for suggestions for both material and references (including sources of homework problems). nominally, the course covers "fundamental group and homology".

i would like to give a higher-categorical perspective on these (and this has been encouraged by my supervisor). in particular, i am not required to carefully prove everything. in my own opinion, that is most important for undergrad classes, where students are still working to understand what mathematics is, what constitutes a proof, etc. by contrast, here the priority is giving exposure and perspective.

among other things, i'd like students to come away understanding the following.
(1) the fundamental groupoid and the equivalence $Cov_X \simeq Fun(\Pi_1(X),Set)$ of categories, for a nice topological space $X$.
(2) the following articulation of eilenberg--steenrod:
(a) the $\infty$-category of finite spaces is freely generated by the point under finite colimits.
(b) the derived $\infty$-category of a ring is stable.
(c) optional: (i) the $\infty$-category of spaces is freely generated by the point under colimits. in particular, finite + filtered = all. (ii) the derived $\infty$-category of …

of course, along the way it will be necessary to introduce the $\infty$-categorical perspective. my current thinking is that this is best done by studying homotopy pushouts and pullbacks in topological spaces, and then the quillen equivalence $sSet \rightleftarrows Top$.

i would expect to take general $\infty$-categorical machinery as a black box; i define an $\infty$-category to be "a DK-equivalence class of $Top$-enriched categories", but emphasize the central role of homotopy-coherence via the definition of quasicategories. see lectures 2-8 here for a past attempt: etale.site/teaching/s19-fact-hlgy

i hope that this will generate some interesting and useful discussion, which i expect to be not without some disagreement. thanks in advance for any comments and suggestions!

Patriot

4:44 PM

You'll invariably leave the students rightfully wondering why we're not talking about spaces in a topology class.

Aaron Mazel-Gee

i should also say that i am a big fan of examples, and i hope to include a good mix of them. i expect to do some during lecture, and to have most of the homework be example-based. at the level i intend to cover the abstract material, i can't think of many good examples that specifically about that material -- though i'd be grateful for suggestions in this direction. so i expect the lectures and homework will be complementary, rather than precisely aligned.

@Patriot oh, i certainly expect to begin with topological spaces! i define a "space" to be a weak equivalence class of topological space, anyways.

i also expect to define cellular and simplicial homology.

Saal Hardali

@AaronMazel-Gee One thing that I found resonated with people with a bit experience in 1-categories that are first exposed to $\infty$-categories is try to explain how "1-categories" is a formal machine that allows one to turn (directed) planar diagrams with points and arrows into proofs. Whereas $\infty$-categories is a formal machine that allows one turn directed diagrams of higher dimensions (polytopes/simplicial complexes) into proofs.

I think emphasizing this perspective at points might ease a bit on this issue of "choosing a model"

Aaron Mazel-Gee

@SaalHardali hey, that's a great slogan! thanks for sharing it, i'd never heard it before.

Saal Hardali

Even going as far as pointing out that even the theory of 1-categories has several different models and that the model with "set of objects" + "set of arrows" + "axioms" isn't god given

@AaronMazel-Gee You're welcome :)

Good luck!

Aaron Mazel-Gee

thanks! i'm quite looking forward to it. i haven't taught in a long time, since i was at MSRI this past spring. i have been realizing that, while research is certainly the reason i do math, teaching provides a much-needed balance.

Patriot

4:50 PM

@AaronMazel-Gee When i was talking about spaces, I didn't mean 'a weak equivalence class of a topological space'. I meant quaternionic spaces, lens spaces, Grassmannians, SO(n), the Poincaré sphere, the Whitehead manifold, Tits buildings, loop spaces, etc.

I think these speak more to their imagination than a DK-equivalence of Top-enriched categories ever could. Frankly, if even you struggle to find applications of $\infty$-categories at their level, then you can't expect the students to see the material and appreciate its value.

Denis Nardin

@Patriot That's why I call my classes "homotopy theory", not "topology". Other people do classes on manifolds

(although next semester I was thinking of covering Thom's theorem on bordism anyway, right after doing Atiyah duality, just because it's a cool one)

Aaron Mazel-Gee

@Patriot oh yes, sorry for being unclear. i know that for most people "space" means "topological space" (and i think this is completely reasonable -- i am a weirdo). what i meant was that topological spaces are still baked into my notion of an $\infty$-groupoid. regarding the list you give, those are some excellent sources of examples, thanks!

regarding the definition of an $\infty$-category, "DK-equivalence class of Top-enriched category" is just meant to be a baseline, so that we are doing mathematics and not philosophy.

Denis Nardin

Just for the record, I do believe in introducing students to Kan complexes early, this independently on whether you want to adopt an ∞-categorical perspective or not

Aaron Mazel-Gee

@Patriot regarding applications of $\infty$-categories, i think that the above articulation of eilenberg--steenrod is a great application. i find it much more satisfying and clarifying than the usual one, with long exact sequences and all that. of course, i will discuss both of them and their relationship.

Denis Nardin

I think there's no point in introducing Δ-complexes and whatnot anymore, and I don't believe that defining Sing X is significantly more complicated than defining C_*(X) (which you have to do in such a class anyway)

Aaron Mazel-Gee

4:59 PM

@DenisNardin cool, i like that idea too. can you elaborate on how kan complexes arise? do you motivate it through the quillen equivalence with topological spaces? i agree that simplicial sets are a great combinatorial gadget, and that no others are necessary in this day and age.

Denis Nardin

@AaronMazel-Gee I can only tell you how I plan to do it next semester, not if it will succeed :)

Aaron Mazel-Gee

haha okay sure, i'm happy to hear that too

Denis Nardin

Essentially it's not hard to prove that Sing |K|→K is a homotopy equivalences if K a Kan complex (although for the sake of time I think I'll blackbox a simplicial approximation lemma and put it only in the notes). After that you can define the homotopy groups of X as those of Sing X. Moreover you can show that X→Y is a Serre fibration iff Sing X→Sing Y is a Kan fibration and that you get a les in homotopy groups etc

Essentially I want to show them that all homotopical information is contained in Sing X. After that I can give them examples of Kan complexes that are not easy to define as Sing of something (e.g. the Kan complex of embedding of manifolds) and this should motivate them enough to accept Kan complexes as "homotopy types"

Ideally I'd like to tell them also about Dold-Kan, but I have to think if I'll have the time for that

If not I'll just define C_* of a Kan complex in the naive way as an alternating sum

Aaron Mazel-Gee

regarding "embeddings of manifolds", do you mean taking isotopies as 1-simplices? is that different from the topological space of embeddings (with an appropriate topology)?

i agree that dold--kan is great, also because then you can easily explain dold--thom: homology is just linearization.

(i also used dold--thom as a warm-up to factorization homology.)

Denis Nardin

5:41 PM

@AaronMazel-Gee It's not, but defining the topological space of embeddings of smooth manifolds is a lot more complicated than people realize... And for example it's not known how to do it for PL manifolds (although students might not appreciate why this is interesting). Comparatively defining the Kan complex is trivial

Saal Hardali

6:47 PM

@AaronMazel-Gee If you're going along what Dennis suggets, I also like the example of C^{infty}(M,N)---->Map(M,N) being a weak homotopy equivalence (for smooth N and compact M) but that LHS isn't obviously a CW complex which motivates why you might care about weak rather than strong homotopy equivalences.

I have no idea how to fit all of this in one course though to be honest ^^

Personally I don't know many examples that motivate weak equivalences rather than strong equivalences other than this, probably there are many though...

When I say motivate I mean not from the point of view of simplicity but rather justifying why weak equivalence are "more correct" than strong.

Adrian Clough

9:12 PM

@AaronMazel-Gee There has been some discussion regarding the difference between homotopy theory and algebraic topology, as well as what is meant by "space". I would consider algebraic topology to be the study of some (oo-)category of geometric objects containing CW-complexes and topological manifolds (classically topological spaces, but one could also consider Delta-generated spaces or sheaves on topological manifolds). These objects have underlying homotopy types;

this is made precise by the nerve functor or by taking the Cech nerve of a good covering. What makes algebraic topology interesting is then the interaction of the geometry and homotopy theory. E.g. a two dimensional closed manifold is completely determined by (the 1-truncation of) its underlying homotopy type.

In the other direction, realising $S^1$ as the standard circle allows for an easy proof of the fact that its fundamental group is $\mathbf{Z}$ (of course, the hardened homotopy theorist might define $S^1$ as $B\mathbf{Z}$... but it could also be defined, say, as $S^0 \star S^0$.).

Another such nice interaction is the following: Vector bundles are a priori geometric objects, but then one notices that a bundle trivialises locally iff it trivialises on contractible subsets of the base; this ultimately explains why vector bundles are controlled by the underlying homotopy type of the base.

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