# What non-categorical applications are there of homotopical algebra?

(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)

More and more lately I have been unexpectedly running into things that might be named 'homotopical algebra' - in glancing at symplectic topology I run into Fukaya categories, and thus of $A_\infty$ categories; in looking at algebraic topology at things above the introductory level I see language like 'model categories' or simplicial sets, or more abstractly, algebraic geometry might lead me to stuff like the work of Jacob Lurie. These things feel bizarre and out-of-place to me - it's all so abstract and dry it's difficult to get a handle on, and feels so removed from ""reality"" (a.k.a., things I have a geometric intuition for).

I understand that this is a result of my maturity level - I'm only just coming to terms with categorical language, and a few years ago I would have probably made a poor joke about "abstract nonsense". Similarly, I have only recently come to love the language of cohomology and sheaves, and this is perhaps because of the problems they solve rather than the language itself; classifying line bundles on a Riemann surface, say, or any of the classical theorems of algebraic topology - both of which can be done elegantly with tools more modern than those questions themselves. I would perhaps feel more at ease if I saw that homotopical algebra can be used for similar purposes, so:

What are some down-to-earth (interesting to non-category theorists, say, that are naturally stated in non-categorical language) questions that have been solved using these sorts of tools, or such that these sorts of tools are expected to be useful? Why is the language of homotopy theory useful to the modern mathematician who is not already immersed in it?

• Thank you for having the guts to ask this. I know many people who might call themselves homotopy theorists (myself included) who feel the same way. – Mark Grant Jun 6 '14 at 7:57
• There's a danger in pushing the 'should solve problems not naturally expressible in category theoretic terms' criterion that you'll resemble someone in 1900 asking what the geometric developments of Riemann, Klein and Lie have done for the down-to-earth study of Euclidean geometry. Your "classifying line bundles on a Riemann surface" became something you now accept as down-to-earth through the visionary ideas of the best mathematicians of the past, not always constrained by the need to resolve the down-to-earth problems of their day. – David Corfield Jun 6 '14 at 14:01
• @DavidCorfield : That kind of rhetoric is part of the reason I stay far, far away from anything having to do with topos theory, $\infty$-categories, etc. – Andy Putman Jun 6 '14 at 15:21
• @ToddTrimble : I don't think he's being aggressive as much as wrong-headed and a little condescending (and his example is way off : classifying line bundles on a Riemann surface is equivalent to studying what kinds of meromorphic functions a Riemann surface admits, which is a very classical thing to study; one of the reasons that Grothendieck-style algebraic geometry became popular was that it gave tools to attack these kinds of old problems). If I ask someone for concrete applications of the ideas they are enthusiastic about and they reply with vague philosophizing and/or (continued) – Andy Putman Jun 6 '14 at 18:48
• suggestions that people who don't adapt their point of view are hidebound reactionaries, then it is hard not to start to suspect that there's not a lot of interesting mathematics there. (nb : I am responding not just to David's comment but also to other discussions of this topic here and elsewhere on the internet). – Andy Putman Jun 6 '14 at 18:50

I like the simple slogan: homotopical algebra is the nonlinear generalization of homological algebra. Let me assume that you value and appreciate homological algebra in the broadest sense as a fundamental, successful and highly applicable tool in many areas of math (otherwise I can't conceive of an argument that would be convincing for this question). At the coarsest level homological algebra is based on the idea of resolutions, i.e. that to perform algebraic operations on objects we should describe them in terms of objects that behave well for the given operations.

Now let's observe that homological algebra is a linear theory, in the sense that it deals with things like vector spaces, modules over a ring, and more generally objects of abelian categories. What if your interests involve more complicated objects that are not linear? for example, rings, algebras, varieties, manifolds, categories etc? philosophically it still makes sense that we have much to gain by resolving in some appropriate sense. Homotopical algebra is the language and toolkit built for this explicit purpose, and with many explicit applications. The $\infty$-language in my mind is just a very convenient and relatively friendly apparatus to understand, navigate and apply this theory.

Some key examples:

$\bullet$ Hodge theory. For me (and I assume many other algebraic geometers) the first instance of homotopical algebraic thinking I encountered was Deligne's construction of mixed Hodge structures on the cohomology of complex algebraic varieties, one of the most powerful tools in modern algebraic geometry. The idea is that the functor "de Rham cohomology" is very wonderfully behaved on smooth complex projective varieties, and most importantly carries a rich extra structure, a pure Hodge structure. We can take advantage of this for say any singular projective variety if we use the idea of resolution, in the form of a simplicial object (a convenient nonlinear version of a chain complex) --- we replace the variety by a simplicial smooth projective variety which is equivalent in the appropriate sense, in particular will produce the same measurement (cohomology). The existence of such is deep geometry (resolution of singularities) but its explicit applications don't require explicit knowledge of this geometry. It now follows that the singular variety's cohomology carries the appropriate derived version of a pure Hodge structure, namely a mixed Hodge structure.

$\bullet$ The tangent complex. Another seminal circa 1970 application is the Quillen-Illusie theory of the tangent complex. Again we want to do basic geometry - this time calculus - on a singular variety, or perhaps let's say a commutative ring, so we resolve it in the sense that befits the problem. We like affine spaces for taking derivatives etc, so if we want to calculate derivatives (tangent spaces) on a singular variety we should resolve it by such --- replace a ring by an appropriate free resolution (this time a COsimplicial variety). This gives us a way to extend the basic tools of calculus to singular varieties, with many corresponding applications.

$\bullet$ The virtual fundamental class. This is an elaboration on the previous point which is much more recent. We would like now to integrate on a class of singular varieties, so need a version of the fundamental class. The varieties in question arise as moduli spaces (say in Gromov-Witten or Donaldson-Thomas theories), which means they are relatively easy to resolve in a natural way (express as a derived moduli problem). As ordinary varieties they are very badly behaved (eg are not even equidimensional) but the derived moduli problem naturally carries a fundamental class.

$\bullet$ In representation theory the key objects of study are again nonlinear --- associative algebras (or equivalently their categories of modules). Thus to perform algebraic operations on these algebras we gain much by allowing ourselves to resolve them. As mentioned above the geometric Langlands program is one place where homotopical language is extremely useful, but one can find the same issues in studying say modular representations of finite groups (eg the theory of support varieties and stable module categories). More generally Hochschild/cyclic theory, the "calculus" of associative algebras/the fundamental invariants of noncommutative geometry, are natural applications of homotopical algebra. There are many spectacular achievements in this area, one famous one being the Deligne conjecture/Kontsevich formality/deformation quantization circle of ideas. The cobordism hypothesis, in my view one of the pinnacles of homotopical algebra, has among its many facets a vast generalization of Hochschild theory.

• I'd have liked to add something intelligent about algebraic K-theory and its triumphs as applications of homotopical algebra but can't pretend to do it justice - though I can link here. – David Ben-Zvi Jun 6 '14 at 23:28
• Nice answer. I was wondering if you could briefly explain the point "The ∞-language in my mind is just a very convenient and relatively friendly apparatus to understand, navigate and apply this theory." Specifically, how does the language of infinity categories make it easier to apply homotopical algebra? – Jason Polak Jun 8 '14 at 1:00
• @JasonPolak: I wrote something of that flavor here: mathoverflow.net/questions/84381/… - the short of it is, $\infty$ categories are perfectly suited to make formal arguments that ought to be formal. It is not a computationally effective language but one that powerfully captures the "pure thought" aspects of algebra: if you're naive about homotopy theory (like me) you can still easily prove analogs of statements that are true for robust structural reasons in a discrete setting, using these tools. A favorite example: cf. Barr-Beck-Lurie theorem. – David Ben-Zvi Jun 9 '14 at 23:07

As a student, I'm always looking for organizing principles in mathematics to help me keep track of all of the mathematics I learn. It's easy to get lost in a deluge of definitions unless I organize them in some way. For a long time category theory was my main organizing principle (e.g. the idea of adjoint functors alone is already a very helpful way to organize constructions in mathematics), but at some point ordinary category theory became inadequate and I needed the language of higher categories (all that nLab stuff).

Here is the sort of thing higher category theory helps me keep organized in my head:

Where do long exact sequences come from? Why was that a thing it should have occurred to us to invent? Where can we expect them to show up in mathematics?

A standard answer is that long exact sequences come from short exact sequences of chain complexes, but this is inadequate for describing at least one very important long exact sequence in mathematics, namely the long exact sequence of a fibration. This is perhaps the first long exact sequence one learns about which involves nonabelian groups, and so cannot come from homological algebra in the usual sense at all. So where does it come from?

From the perspective of higher category theory, long exact sequences are shadows of two dual and more fundamental constructions, namely fiber sequences and cofiber sequences. These in turn come from repeatedly taking homotopy pullbacks resp. homotopy pushouts, which one can think of as the "nonabelian derived functors" of ordinary pullbacks resp. pushouts (which are not homotopically well-behaved and must be corrected). The long exact sequence of a fibration in particular comes from a fiber sequence of the form

$$\cdots \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B$$

where $F \to E \to B$ is a fibration, whereas the long exact sequences in ordinary homology and cohomology come from a cofiber sequence of the form

$$A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma X/A \to \cdots$$

where $A \to X$ is a cofibration. The higher categorical point of view tells you at least one interesting thing right off the bat, which is that these two constructions are categorically dual to each other: running the first one in the opposite category gets you the second one! (And if there's one thing any mathematician should respect, it's a duality.)

More generally, there's no reason to restrict our attention to spaces: the higher categorical machinery runs in any higher category with the right structure, and in particular running it in chain complexes also gets us the long exact sequence associated to a short exact sequence of chain complexes, while also explaining that the homological mapping cone is not only analogous to but is precisely the same construction as the topological mapping cone: they're both homotopy pushouts.

Let me also make some other comments.

What's the deal with model categories and simplicial sets?

You say that you've come to love the language of cohomology and sheaves. Great: if you're happy with the idea of using chain complexes as resolutions of objects to compute things like cohomology, then the main thing to know about the model category story is that

• model categories are a setting for understanding and computing with "nonabelian resolutions" (in particular to make sense of "nonabelian derived functors"), and
• simplicial objects can be used to build these resolutions; in particular, they can be thought of as "nonabelian chain complexes."

That second claim can be made precise using the Dold-Kan theorem, which tells you that the category of simplicial objects in an abelian category is equivalent to the category of chain complexes concentrated in nonnegative degree.

Here's a relatively concrete example. The Cech nerve of a nice cover $U \to X$ of a space is a simplicial object which resolves the space $X$ in a particular sense; in particular, if the cover has the property that every finite intersection of opens in the cover is contractible, the resulting resolution can be thought of as a nonabelian analogue of a free resolution of a module. The abelian version of this story, where you're mapping $X$ into abelian objects like Eilenberg-MacLane spaces, gives you Cech cohomology, but the nonabelian version of this story gives you, for example, the Cech cocycle description of a principal $G$-bundle.

The Cech cocycle description of a principal $G$-bundle is a great place to start seeing higher category theory at work. First, let me recall the following: if $U_{\alpha}$ is an open cover of a space $X$, then to specify a continuous function $f : X \to Y$ is precisely the same data as specifying

• continuous functions $f_{\alpha} : U_{\alpha} \to Y$ for all $\alpha$
• having the property that the restrictions of $f_{\alpha}$ and $f_{\beta}$ to their intersection $U_{\alpha \beta} = U_{\alpha} \cap U_{\beta}$ agree.

This is a Cech $0$-cycle description of $\text{Hom}(X, Y)$; we are just using the fact that $X$ is the coequalizer of a certain diagram built out of the $U_{\alpha}$s.

Why doesn't this suffice to describe principal bundles? It's because the functor $\text{Hom}(-, Y)$ preserves colimits, but the functor $[-, BG]$ doesn't, because we're taking homotopy classes. Another way to say this is that there is really a groupoid of principal $G$-bundles on a space $X$ (the fundamental groupoid of the mapping space $\text{Maps}(X, BG)$, in fact) and we want to know this groupoid, or at least its set of isomorphism classes.

The functor $[-, BG]$ doesn't preserve colimits, morally because taking colimits isn't guaranteed to play nicely with taking homotopy classes. However, it does play nicely with homotopy colimits, and the precise sense in which a Cech nerve of a space $X$ is a resolution of a space is that, under nice hypotheses, $X$ is the homotopy colimit of that Cech nerve. You can think of this as a fancier version of the coequalizer we talked about above, which is why to specify a principal $G$-bundle you need to talk about triple intersections instead of double intersections: you need

• continuous functions $g_{\alpha \beta} : U_{\alpha \beta} \to G$ for all $\alpha, \beta$
• having the property that the restrictions of $g_{\alpha \beta}, g_{\beta \gamma}, g_{\alpha \gamma}$ to their common intersection $U_{\alpha \beta \gamma} = U_{\alpha} \cap U_{\beta} \cap U_{\gamma}$ satisfy the cocycle relation $g_{\alpha \beta} g_{\beta \gamma} = g_{\alpha \gamma}$.

This is precisely a morphism between truncations of the simplicial objects, or nonabelian resolutions, given on the one hand by the Cech nerve of the cover $U \to X$ and on the other hand by the bar resolution of $BG$!

Thinking about algebraic topology this way has made it more topological for me: the above story can be adapted to explain ordinary Cech cohomology in a way that doesn't involve passing to chain complexes at any step, for example (the fact that you can in fact use chain complexes comes from Dold-Kan), and more generally doing algebraic topology this way lets you replace homological constructions with constructions that are genuinely about spaces.

• Wow! Great, great answer. – Filippo Alberto Edoardo Jun 7 '14 at 14:47
• Hi Qiachou, Lately I find myself reading a lot of your posts and find them really enlightening. What would you rcommend to a novice like myself who relates philosophically with the homotopical(-higher categorical) viewpoint but lacks a deep understanding of the foundations of homotopy theory? Is there a "Modern Homotopy theory demystified" book you can recommend? – Saal Hardali Nov 11 '15 at 18:03
• @Saal: try Strom's Modern Classical Homotopy Theory. I haven't read it but it at least bothers to talk about fiber and cofiber sequences, which is promising. – Qiaochu Yuan Nov 11 '15 at 19:27
• @QiaochuYuan Thanks for the recommandation! May i ask how did you come to learn all this stuff? – Saal Hardali Nov 11 '15 at 19:31
• @Saal: I read stuff on the nLab and by John Baez (particularly math.ucr.edu/home/baez/cohomology.pdf), thought about stuff, worked through some examples... – Qiaochu Yuan Nov 11 '15 at 19:33

As already mentioned, the phrase "down to earth" is subjective. For instance, I have friends in analytic number theory who ask what the utility of categories is (not higher categories), and to some, they might sound like a mathematician asking what the utility of groups is, but this is simply a reflection of our backgrounds.

Regardless, here's a brief list of ways in which homotopical algebra'' has appeared in, helped organized, or has helped advance, mathematics. Perhaps one of these is an application you'll find down to earth.

1. Making algebraic geometry even more geometric (ex: Homological algebra). This actually is a motivation for homological algebra, more than anything.

We all know that to count the "right" intersection number between two shapes, we should wiggle them into general position and then count things. Unfortunately, in algebraic geometry, you can't always wiggle things. (For instance, over a field of finite characteristic.) But Serre's intersection formula--which replaces the usual tensor product with Tor--tells you that you can get the "right" answer without wiggling, and rather by deriving into the world of chain complexes. So suddenly, the ring of functions of an intersection is not just an algebra, but a dg algebra. Enter homotopical algebra.

Thus to make intuitions from ordinary geometry fit into algebraic geometry, we see inevitably the need for algebras in chain complexes. It's still mysterious that chain complexes should be so useful for capturing this geometry, but it works: Things like dg schemes, derived algebraic geometry a la Toen-Vezzosi, and deformation theory, all need the language of homotopical algebra to capture the geometry of algebraic geometry.

For instance, here's an application in the last example: A "deformation problem," in a nutshell, is a way to start off with a thing and find an infinitesimal family of such things--i.e., to find a "tangent vector" in a space of things. Lurie talked about this in his ICM talk. With the right language for homotopical algebra, you can actually prove a theorem stating that every deformation problem in characteristic zero is encoded by an $L_\infty$ algebra (i.e., a homotopical version of a Lie algebra). I think this is a beautiful way to make concrete the intuition of a deformation problem encoding the tangent space to something.

As an early application predating this theorem, Goldman and Millson could say something about what singularities on the moduli space of flat G bundles on a Kahler manifold looks like, by studying cohomological properties of an associated dg Lie algebra.

2. Giving geometric meaning to algebraic structures (ex: Factorization homology). This might sound a lot like the above application, but this one is more about manifolds than about algebraic geometry.

There are famous animals with interesting algebraic structures: String topology gives rise to BV algebras, Hochschild cohomology of an associative algebra is a Gerstenhaber algebra, et cetera. These are some "first examples" that illustrate the need for some language that can uniformly describe interesting algebraic structures. For instance, why is there a Connes B operator on Hochschild homology, and what is the geometric meaning behind the HKR theorem, which states that Hochschild homology of a smooth algebra is the algebraic differential forms? Well, geometry (and geometry of manifolds) has a way of organizing (or fooling us into think we've organized) a lot of data, so can we think about these algebraic structures geometrically? (String topology being the most obviously geometric example.)

If so, manifolds are things that can wiggle around, and embeddings between them can wiggle around. So can we transport the homotopical content of manifold theory to the algebraic realm? You can, so long as you develop a good language of homotopical algebra. For instance, just realizing that both manifolds and chain complexes form an oo-category, you can try to express some of the invariants above as certain functors from the oo-category of manifolds to the oo-category of chain complexes. This is a concise and concrete way to try to explain things like: Hochschild homology is what you assign to the circle, and the circle has a circle action on it, so Hochschild homology has a circle action, too.

I'm a bit biased because I think about factorization homology pretty often, but I think this is one great example of an application of homotopical algebra. For instance, you can recover quantum link invariants (work of Costello-Francis) and interpret Koszul duality via Poincare duality (work of Ayala-Francis). It's a beautiful tool for injecting geometry into algebra, and I think that's an application of homotopical algebra that doesn't require categorical language to convey.

3. Evolving from triangulated categories to dg categories. This doesn't fully fit the bill of the OP's prompt, because it requires category theory to state the problem. Regardless, I think it's a great example of departing from algebra to homotopical algebra. As is well-known, mapping cones are not functorial in the triangulated category setting. So triangulated categories do not form a natural language for capturing relationships between categories of complexes of things. (DbCoh(X), Chain, etc.) On the other hand, pre-triangulated dg categories do the job; I think this is the most accessible victory of homotopical algebra over non-homotopical algebra: By remembering the entire chain complex of morphisms (i.e., all homotopies and higher homotopies) you have enabled yourself to play categorically with a very natural construction: Taking cokernels and kernels (i.e., taking mapping cones and their shifts).

4. Organizing structures. This is a meta-application, because I think all of algebra is meant to somehow organize structures. Theory for the sake of theory is not a bad thing, because theory helps us organize seemingly disparate ideas, like how to compare a category of sheaves with a category made up of Lagrangians and holomorphic disk-counting (mirror symmetry), or characterizing n-fold loop spaces as group-like E_n algebras (May's recognition principle), or recognizing when a certain category is a category of modules (souped-up version of Freyd-Mitchell embedding theorem to the homotopical setting).

5. It just pops up. This may in some ways be the least illuminating reason, but I'd like to make a remark on Fukaya categories. From the point of view of an algebraist, the language of Aoo categories does not emerge in Fukaya categories out of a purely algebraic necessity; I would rather say that the Aoo world rears its heads in this setting because of the following fact: The moduli spaces of holomorphic structures on polygons is diffeomorphic to the Stasheff polyhedra. (Stasheff polyhedra organize one model for the Aoo operad, and it's the model most commonly used to characterize Aoo categories.)

So, somehow, the spirit with which Aoo language emerges in the Fukaya world is not to blindly capture an unspecified, fuzzy, derived algebra; rather, Aoo language is somehow the natural road forced upon you by the geometry of holomorphic polygons.

Of course, this probably helped popularize Aoo categories, and spurred a lot of generalizations of dg constructions (triangulated envelopes, constructing an Aoo category of Aoo categories, et cetera); but I wanted to remark that the reason that Aoo language pops up can be seen as a purely geometric one. In trying to enhance the cohomology-level Fukaya constructions (e.g., the Donaldson category), one might have tried to find a dg enhancement (which is already homotopical algebra) but one inevitably runs into the Aoo world via geometry.

As soon as you do serious homotopy theory in a context outside topological spaces, a formalism for abstract homotopy theory is very helpful. Let's give a few examples:

1) Waldhausen's algebraic K-theory of spaces. This is used in a number of contexts for high-dimensional manifolds, e.g. for studying the homotopy type of their diffeomorphism groups. But mainly I want to quote a paragraph from the introduction:

The first part of the paper, on which everything else depends, may perhaps look a little frightening because of the abstract language that it uses throughout. This is unfortunate, but there is no way out. It is not the purpose of the abstract language to strive for great generality. The purpose is rather to simplify proofs, and indeed to make some proofs understandable at all. The reader is invited to run the following test: take Theorem 2.2.1 (this is about the worst case), translate the complete proof into not using the abstract language, and then try to communicate it to somebody else.

2) Modern stable homotopy theory depends on good model categories (or $\infty$-categories) of spectra. Arguments get very clumsy if one wants to do stable homotopy theory without the homotopy category of spectra at all, but for some of the modern things one needs even something better than the homotopy category. Two prominent examples:

a) The Goerss-Hopkins obstruction theory. This combines good model categories of spectra with good model categories of resolutions (similar to the ones for Andre-Quillen homology). This is important to construct new examples of spectra/cohomology theories, e.g. higher real K-theory and topological modular forms. These have been used to construct new infinite families in the stable homotopy groups of spheres (and with that: new infinite families of exotic differentable structures on spheres).

b) The Kervaire invariant one problem: The original problem can be phrased as: Is every framed manifold framed bordant to an (exotic) sphere? It turns out that the answer is yes in dimensions $\neq 2^n-2$. Hill, Hopkins and Ravenel proved in 2009 that the answer is also yes in all dimensions $\geq 254$. They rather proved an equivalent formulation of this in stable homotopy theory. For this they used some fairly serious equivariant homotopy theory. That they provide a 34-page appendix on model structures on equivariant spectra should be proof enough that this language is useful for answering this very geometric question!

3) There is hidden (abstract) homotopy theory in many disciplines. For example in geometric group theory you here people say that a finite index subgroup is virtually (or also: more or less) isomorphic to the whole group. One can just declare these inclusions to be weak equivalences and get a category with weak equivalences (aka a relative category). In its homotopy category, the automorphism group of an object is exactly a commensurator, i.e. an isomorphism from one finite index subgroup to another, up to some equivalence. Likewise in algebraic geometry if one declares inclusions of dense open subsets as weak equivalences. One should get as morphisms in the homotopy category exactly rational maps and as isomorphisms birational equivalences. As soon as we say that two things are more or less the same, but not really isomorphic, we are doing secretly a kind of homotopy theory.

• "As soon as we say that two things are more or less the same, but not really isomorphic, we are doing secretly a kind of homotopy theory." This sounds like a sort of homotopy-theory imperialism, to me. Could you give an actual example where the tools of homotopy theory are useful in studying commensurability classes of groups, say? – HJRW Jun 7 '14 at 18:35
• No, not in this form. But where do tools from category theory help to study the category of topological spaces or of commutative algebras? It is more that category theory provides common proofs of some (simple) lemmas and works as an organizing principle, which might suggest useful viewpoints. Likewise, as soon as one postulates that finite index subgroups are more or less the same as the whole group, homotopy theory suggests to look at commensurators (as isomorphisms in the homotopy category)... – Lennart Meier Jun 7 '14 at 20:34
• ...Homotopy theory also suggests to look at mapping simplicial sets between groups (which might or not be useful - I am no expert). But do not hope for great revelations. In this generality, homotopy theory is rather formal. – Lennart Meier Jun 7 '14 at 20:36
• '[W]here do tools from category theory help to study the category of topological spaces or of commutative algebras?' Well, the whole point of this thread was to give examples where the homotopy-theoretic viewpoint is useful! And some of the answers, including your own, are very interesting. On the other hand, as has been pointed out elsewhere, some of the rhetoric about homotopy theory is rather grandiose; I'm afraid I think your point 3 is an example of this. – HJRW Jun 8 '14 at 1:20

Goodwille calculus (a.k.a. calculus of functors) was motivated (at least in part) by the study of embeddings of manifolds. In one (probably standard) introduction, Goodwille pointed out that the subject of knot theory is the study of the connected components of the space of embeddings $S^1 \rightarrow S^3$.

From this point of view, maybe the study of $\pi_* \mathrm{Emb}(S^1, S^3)$ for $* > 0$ seems reasonably "down to earth"? The calculus of functors gives an approach to this problem by decomposing the embedding functor into "derivatives".

In any case, I think this example illustrates a more general point that "homotopical algebra" generally has to do with studying the higher-order homotopy groups of things, and these are generally interesting when you want to know something like the number of equivalence classes of equivalences between things.

• It's quite unclear what the Goodwillie calculus sees of the homotopy-groups of $Emb(S^1,S^3)$. In contrast, it would appear to see "quite a bit" of the homology. But in homotopy, it looks like the calculus sees relatively little. For example, it would be very difficult to extract much of the non-abelian nature of $\pi_1 Emb(S^1,S^3)$ using the calculus. In higher dimensions $Emb(S^1,S^n)$ for $n > 3$ all those troubles vanish, though. – Ryan Budney Apr 2 '15 at 22:14
• Also, the higher homotopy groups of $Emb(S^1,S^3)$, with the exception of the unknot component, are the same as the higher homotopy groups of $S^3$. I don't think the embedding calculus offers much insights into that -- from this perspective the homotopy groups of spheres are building blocks rather than things to gain insight into. – Ryan Budney Apr 2 '15 at 22:21

You mention an interest in symplectic geometry and Fukaya categories. A very natural problem in the field is given, say, a Liouville domain (a very reasonable class of exact symplectic manifolds) what are the diffeomorphism types of exact Lagrangian submanifolds. The strongest results on these sorts of problems are all obtained with a heavy dose of homological or homotopical algebra.

In the simplest case of a cotangent bundle of a compact oriented manifold, you can examine Abouzaid's series of papers culminating in:

http://arxiv.org/abs/1005.0358

In later work with Smith, they used similar homological algebra (as well as some very interesting geometry) to classify Lagrangians in certain plumbings of cotangent bundles:

http://arxiv.org/abs/1107.0129

Seidel has also done some very spectacular work, by studying a homotopical analogue of a dilating $\mathbb{C}^*$ action on an $A_{\infty}$-category.

Finally there is work by Fukaya (studying a weaker condition than exact) focused on Lagrangains inside of $\mathbb{C}^n$ in a paper called "Applications of Floer Homology of Lagrangian Submanifolds to Symplectic Topology". This uses ideas from operads and string topology.

Higher algebra and higher geometry are like air: they are everywhere -- even if often you may only notice it when a storm comes up. It's the sea rising:

I can illustrate the second approach with the same image of a nut to be opened. The first analogy which came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months – when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.

(quote form Alexander Grothendieck, Récoltes et semailles, 1985–1987, pp. 552-3-1)

I don’t want you to think all this is theory for the sake of it, or rather for the sake of itself. It’s theory for the sake of other theory.

(quote from Jacob Lurie, ICM talk 2010)

I have started to compile some commented lists of applications of higher algebra and higher geometry in these nLab entries:

These entries all could and eventually deserve to be expanded much more. The limiting factor is not the number and scope of examples. If you ask for something more specific, maybe it inspires me (or somebody reading this here) to go and add more to these nLab entries.

I don't have a non-categorical application of all the topics you discuss but I can give you one.

There is a topological operad called "the splicing operad" that acts on embedding spaces. It acts on a space closely related to the space of smooth embeddings of $S^1$ in $S^3$, I call this space $\mathcal K_{3,1}$, or the space of `long embeddings' of the real numbers in Euclidean 3-space. It turns out this operad is closely related with many of the biggest structure theorems in 3-manifold topology. Namely,

The splicing operad (in dimension 3) is a free product of a certain free operad and $O_2$ semi-direct product the $2$-cubes operad. Moreover, the knot space $\mathcal K_{3,1}$ is freely generated by the splicing operad and the subspace of $\mathcal K_{3,1}$ consisting of torus and hyperbolic knots.

This theorem, on the level of the path-components of $\mathcal K_{3,1}$ is precisely a description of the geometrization theorem, restricted to the class of knot exteriors. For the higher homotopy groups the 3-manifold content of this theorem consists of the Smale conjecture, together with the equivariant connect-sum and torus decompositions of 3-manifolds, and the theorem that the actions of finite groups on $S^3$ are conjugate to linear actions.

So this operad sees quite deeply into 3-manifold topology. There are analogous operads in higher dimensions, and also operads of this type for spaces of string links (due to Burke and Koytcheff). What these operads say about embeddings and diffeomorphisms groups of high-dimensional manifolds is an open problem. High-dimensional diffeomorphism groups are presently little-understood.

I can't resist posting my latest publication as an answer: Koenig, Külshammer, Ovsienko: Quasi-hereditary algebras, exact Borel subalgebras, A-infinity categories and boxes.

Let me sketch the result, which is exactly of the kind: You have a problem that is stated purely in down-to-earth language and the only known solution requires higher categorical language, in this case A-infinity categories and boxes.

The motivation comes from the classical example that in the representation theory of semisimple complex Lie algebras, one often studies induced representations. The universal highest weight modules, the Verma modules, for $U(\mathfrak{g})$ are induced from the simple modules for the subalgebra $U(\mathfrak{b})$, where $\mathfrak{b}$ is a Borel Lie subalgebra of $\mathfrak{g}$. The PBW theorem implies that $U(\mathfrak{g})$ is free over $U(\mathfrak{b})$, thus tensoring gives an exact functor $U(\mathfrak{b})-\operatorname{mod}\to U(\mathfrak{g})-\operatorname{mod}$ and one can study parts of the representation theory of $U(\mathfrak{g})$ by understanding the representation theory of $U(\mathfrak{b})$.

Now, the formulation of the problem is the analogue of this problem in the world of finite dimensional algebras. The analogue of $U(\mathfrak{g})$ is called a quasi-hereditary algebras. Roughly speaking, these are algebras having certain modules, called standard modules, which behave similarly to Verma modules for $U(\mathfrak{g})$. An "obvious" question is whether each quasi-hereditary algebra $A$ also possesses the analogue of a $U(\mathfrak{b})$, which is called an exact Borel subalgebra $B$. Here, the standard modules should be induced from the simple modules for $B$ and $A$ should be projective (slight generalisation of free) over $B$. This again implies that the induction functor, $B-\operatorname{mod}\to A-\operatorname{mod}$ is exact. Additionally, $B$ should satify an analogous property to solvability of $\mathfrak{b}$, called directedness. This formulation definitely no higher categorical methods and almost no category theory.

The answer to the question is no, if you want to stick to your algebra $A$. This is known since 1995. But if you allow to change from $A$ to an algebra which has the same representation theory, i.e. an algebra $R$ Morita equivalent to $A$, the answer is yes. So, also the answer to the problem requires no higher categorical methods.

In contrast, our proof does. We start by considering the Yoneda Ext-algebra of the standard modules, regarded as an $A_\infty$-category. We translate this to an algebra $B$ together with a $B$-coalgebra $W$ (this is what is sometimes called a box). Then the dual algebra of the coalgebra $W$ does the job. To prove that it does, in each translation step we also use different descriptions of the category of modules filtered by standard modules, i.e. twisted modules for $A_\infty$-categories, modules over a box.

The nonabelian tensor product of groups which act on each other (in a compatible way) was defined by Jean-Louis Loday and myself in 1984 and the topic has been taken up by group theorists because of its relation to commutator theory. I have compiled a bibiliography on this topic which now runs to 131 items. The applications to homotopy theory do require tools of higher groupoids for the proofs, as well as advanced techniques from algebraic topology, and many useful aspects of category theory.

Just to give an idea of the subject, suppose $M,N$ are normal subgroups of a group $P$. Then the commutator map $[\;,\;]: M \times N \to P$ is not bimultiplicative, but is what may be called a biderivation. So it factors through a universal biderivation $M \times N \to M \otimes N$ giving rise to a morphism $\kappa: M \otimes N \to P$. Even the calculation of this tensor product is not so easy.