36
$\begingroup$

As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been a narrowing of focus, via a choice to ignore certain "pathological"—or at least intractably complicated—phenomena; instead of considering all topological spaces, one focuses only on compactly generated spaces or CW complexes or something. Or maybe one chooses to focus only on stable homotopy groups. Other times, there seems to have been a broadening of perspective, as new objects of study are introduced to fill perceived gaps in the landscape. Spectra are one notable example. I was fascinated when I discovered the 1991 paper by Lewis, Is there a convenient category of spectra?, showing that a certain list of seemingly desirable properties cannot be simultaneously satisfied. More recent concepts include model categories, $\infty$-categories, and homotopy type theory.

I was wondering if someone could sketch a timeline of the most important such "foundational shifts" in homotopy theory over the past 50 years, together with a couple of brief sentences about what motivated the shifts. Such a bird's-eye sketch would, I think, help mathematicians from neighboring fields get some sense of the purpose of all the seemingly high-falutin' modern abstractions, and reduce the impenetrability of the current literature.

$\endgroup$
1

3 Answers 3

40
$\begingroup$

Such a timeline is necessarily highly subjective.

With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory. The list below concentrates on developments that in some way affect the foundations of homotopy theory, as opposed to general advances in homotopy theory. Given the length of the list, I probably omitted many important developments, feel free to point them out in the comments! I also excluded from consideration the last decade or so, restricting to older developments.

Poincaré defined homology (via Betti numbers) and the fundamental group in a series of papers starting from 1895. The initial approach was nonrigorous, but in response to the resulting criticism, Poincaré reformulated his work in terms of simplicial complexes.

Fréchet defined metric spaces in 1906 and Hausdorff defined topological spaces in 1914. This enabled to study the topological properties of spaces without first triangulating them.

Around 1925, Emmy Noether proposed to upgrade Betti numbers to homology groups. In connection with this, sometime in 1930s, the terminology shifted from “combinatorial topology” to “algebraic topology”.

Around 1931, Veblen and J. H. C. Whitehead introduced the modern definition of a smooth manifold.

Eilenberg defined singular homology in 1943, which resulted in a systematic study of homology and cohomology (defined by Kolmogoroff and Alexander in 1936) of arbitrary topological spaces.

Around 1945, Leray introduced sheaves and spectral sequences. The relevant theory was further developed by Cartan, Serre, and others.

Eilenberg and MacLane introduced categories, functors, and natural transformations in 1945. Ever since then, category theory played an increasingly important role in homotopy theory, to the point where we are now often unable to cleanly separate them.

Eilenberg and Zilber developed the theory of simplicial sets (known at the time as “complete semi-simplicial complexes**) in 1949.

J. H. C. Whitehead proved what is now known as the Whitehead theorem in 1948.

Eilenberg and Steenrod published their Foundations of Algebraic Topology in 1952, formulating what is now known as the Eilenberg–Steenrod axioms.

Around 1953, Cartan and Eilenberg completed their book on homological algebra (published in 1956).

Kan (advised by Eilenberg) systematically developed simplicial homotopy theory (and briefly also cubical homotopy theory) starting from around 1955. He introduced combinatorial homotopy groups, the Dold–Kan correspondence, adjoint functors, limits and colimits, Kan extensions, etc.

Lima defined spectra in 1958.

Quillen published his Homotopical Algebra in 1967, introducing model categories and using them in his Rational homotopy theory around 1968. Around 1972, he introduced higher algebraic K-theory.

In 1971, Gabriel and Ulmer published their systematic account of locally presentable categories.

Segal introduced Γ-spaces around 1972. At the same time, May introduced operads, also in connection with infinite loop spaces.

Brown studied the homotopy theory of sheaves of spaces and spectra in 1972.

Boardman and Vogt introduced quasicategories in 1973.

In 1977, Sullivan published his work on rational homotopy theory in the language of commutative differential graded algebras, complementing the previous work by Quillen.

Dwyer and Kan introduced and developed the theory of simplicial localizations starting from around 1979.

Around 1979, Bousfield introduced what is now known as Bousfield localizations.

In 1983, Grothendieck introduced what is now known as Grothendieck homotopy theory, as well as derivators.

In 1980s, Joyal established what is now known as the Joyal model structure on simplicial sets.

In mid-1980s, Segal (following Witten) introduced what is now known as functorial field theory, later studied by Atiyah, Kontsevich, Freed, Lawrence, and many others.

In 1985, Jardine gave an account of simplicial presheaves.

Around 1986, Lewis, May, Steinberger, McClure introduced genuine equivariant spectra.

In 1989, Makkai and Paré published a systematic account of accessible categories.

In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture about (∞,n)-categories for arbitrary n.

In 1997, Elmendorf, Kriz, Mandell, May published the first ever account of a symmetric monoidal category of spectra.

In 1998, Hovey, Shipley, Smith published an account of symmetric spectra.

In 1998, Rezk introduced complete Segal spaces.

In the late 1990s, Voevodsky introduced and developed motivic homotopy theory (including some joint work with Morel).

Around the late 1990s, Smith introduced combinatorial model categories and proved what is now known as the Smith recognition theorem and established the existence of left Bousfield localizations of left proper combinatorial model categories.

Monoidal model categories were systematically studied by Schwede and Shipley starting from 1997.

In 2006 (based on a 2003 preprint), Lurie's Higher Topos Theory came out, first as an online draft, which was later published.

$\endgroup$
10
  • $\begingroup$ Wow, thanks! I've added links for the past 50 years. If there are better links, please edit accordingly. $\endgroup$ Jun 17 at 2:06
  • 4
    $\begingroup$ Many thanks for your answer, but (for completeness) I recommend you to include "Pursuing stacks" where (among many other things) started higher categories (the homotopy hypothesis)... $\endgroup$ Jun 17 at 2:43
  • $\begingroup$ Oh wow, I would not have guessed the right decade for Boardman-Vogt. $\endgroup$ Jun 17 at 2:52
  • 1
    $\begingroup$ @user234212323: Added a paragraph about Grothendieck's work and a few other items. $\endgroup$ Jun 17 at 15:58
  • 2
    $\begingroup$ surely Pontryagin and Thom deserve shoutouts $\endgroup$ Jun 17 at 22:10
19
$\begingroup$

I think Dmitri Pavlov does a nice job laying out a timeline. Instead of writing a competing answer, let me just try to add in a couple of things I think he left out, addressing a few points from the OP.

1940s Leray invents spectral sequences (while imprisoned by the Nazis!) This becomes a critical computational tool in homotopy theory.

1948-1949 Whitehead introduces CW complexes (and then proves Whitehead's Theorem like Dmitri wrote)

1950 Serre introduces spectral sequences to homotopy theory, and they quickly become one of our strongest computational tools.

1962 Brown proves what is now known as the Brown representability theorem. Adams proves another important version in 1971. This work allows one to represent a functor $F$ by an object $E$, meaning $F(X) \simeq Hom(X,E)$. There are various versions depending on how you interpret $\simeq$ and $Hom$. Later, this work lets homotopy theorists see the connection between spectra (objects) and generalized cohomology theories (functors).

1967 Steenrod gives a list of conditions any category of spaces should satisfy to be called a convenient category of spaces. For one thing, they must be Cartesian closed. The category of compactly generated spaces is convenient, as is the category of compactly generated weak Hausdorff spaces introduced by McCord in 1969 because it has even better properties. The category of CW spaces is "too small", and that's part of the reason simplicial sets gained popularity.

1969-1970: May introduces operads (after closely related concepts used by Boardman, Vogt, and Kelly) and uses them for infinite loop space machines. Homotopy theory expands to touch universal algebra.

1983: Grothendieck introduces the homotopy hypothesis as part of Pursuing Stacks, and this guides the deepening connection between homotopy theory and (higher) category theory.

1984 Ravenel formulates the Ravenel conjectures to guide chromatic homotopy theory. This field develops computational tools (to compute stable homotopy groups and other generalized cohomology groups) as well as abstract advances like the Landweber Exact Functor Theorem.

1988 Devinatz, Hopkins, and Smith prove the Ravenel conjectures, except for the Telescope Conjecture, which remains open to this day. Homotopy theory begins to grow more categorical and more abstract approaches gain traction.

1990: Goodwillie introduces functor calculus, spawning a new branch of homotopy theory. This gives another powerful computational tool.

1990s: I would add Hovey to the list of people who systematically studied monoidal model categories. He introduced them, for example His book and his preprint monoidal model categories worked out important aspects of the theory. Hovey also introduced abelian model categories, by which homotopy theory can touch homological algebra (ok, Quillen did this too) and representation theory (via the stable module category).

1994: Adamek and Rosicky work out the theory of locally presentable categories, which is foundational to combinatorial model categories and presentable $\infty$-categories, and makes possible many of the advances in Lurie's books in the 2000s.

1998: Voevodsky introduces motivic homotopy theory, and uses it to prove the Milnor conjecture. He wins a Fields Medal in 2002. Homotopy theory expands to touch algebraic geometry.

2001: Mandell, May, Schwede, and Shipley prove that the various models of spectra are monoidally equivalent in a strong way, hence you can use any of them that you want to.

2005 or so: Jeff Smith introduces the category of Delta-generated spaces, but never publishes anything about it. Others prove it's convenient and has a combinatorial model structure. All choices of categories of spaces mentioned are equivalent homotopically (I mean Quillen equivalent).

2000s: many different models are introduced for the notion of an $(\infty,1)$-category, then proven to be equivalent. Same story subsequently for $(\infty,n)$-categories.

2009: Voevodsky, Awodey, Warren, Shulman, and others introduce homotopy type theory. Homotopy theory expands to touch type theory, logic, and computer science. Part of the motivation is to improve the foundations of all mathematics and to develop workable proof checking software.

2010s: Lack, Verity, Riehl, Garner, Bourke, and many others develop homotopy theory for 2-categories and enriched categories, as well as the $\infty$-cosmoi approach to $\infty$-category theory.

$\endgroup$
13
  • 4
    $\begingroup$ The OP wrote "Other times, there seems to have been a broadening of perspective, as new objects of study are introduced to fill perceived gaps in the landscape. Spectra are one notable example." It's worth pointing out that the gap spectra filled was to have a geometric/topological object to represent stable homotopy groups which were, till then, purely algebraic. $\endgroup$ Jun 17 at 7:03
  • 3
    $\begingroup$ Actually, locally presentable categories date back to at least the 1970s (Gabriel and Ulmer) and were probably known in some embryonic form to the Grothendieck school even earlier. $\endgroup$
    – Zhen Lin
    Jun 17 at 11:16
  • 2
    $\begingroup$ The claim about monoidal model categories is incorrect. Monoidal model categories were introduced by Schwede and Shipley in their January 1998 preprint. They do claim the original credit for this concept in their paper, and there are no preceding works by other authors where this concept is introduced. $\endgroup$ Jun 17 at 16:05
  • 2
    $\begingroup$ @DavidWhite: As Hovey himself writes in his paper: "The first results on this subject were obtained in [SS97]. This paper is a followup to that paper.", where SS97 is the Schwede–Shipley paper. Conversely, Schwede–Shipley do not attribute their definitions to Hovey. Of course, the pushout product axiom was already known to Quillen and both authors point this out. $\endgroup$ Jun 17 at 21:14
  • 4
    $\begingroup$ I figured out the confusion. There are two different 1998 versions of the Schwede-Shipley paper. Mine was dated December 4 and cites Hovey's book, which was written during the 1997-1998 academic year, hence available in the spring of 1998 to his friends. The arxiv version of the Schwede-Shipley paper is dated January 19, 1998. I never saw it till now. So, I think Dmitri is right, and I've edited my answer to not credit Hovey with the introduction of monoidal model categories. $\endgroup$ Jun 21 at 5:27
5
$\begingroup$

Thanks to both Pavlov and White there is now an almost complete list of "critical points" in the history of homotopy theory.

There are a few items that perhaps should make it into the list, for instance the Archaic Period: Betti introduced Betti's numbers here in 1870 -incidentally, as simple as they are, Betti's numbers continue to play a dramatic role in applied math, for instance in Persistent Homology.

I would suggest one tiny emendation: extend your 50 years slightly, by five years, to include a true Annus Mirabilis (1967), namely the work of Quillen on Model Categories.

Let me say why I feel this is indeed a breakthrough and the beginning of modern era (some of the work after 2005 is properly speaking not modern, but post-modern, see below).

Before Quillen, in 1952 Steenrod and Eilenberg managed to get the Grand Unification of Cohomology, certainly a major breakthrough in the series of "foundational efforts" in Algebraic Topology.

In a way Quillen tried to do the same for homotopical algebra, by introducing a sets of axioms for "doing homotopy " in a category. The key notion here is weak equivalence, ie a sets of maps in the ambient cat which contain all the isomorphisms.

This simple step is a foundational paradigm change, because it tells us WHAT Homotopy is all about:

we move from equality (set theory)_ to isomorphism (category theory) to equivalence (homotopy theory).

Quillen add some axioms on formal fibrations, cofibrations, to compute the so-called homotopy limits and colimits, ie lims and colims "up to homotopy"

NOTE before quillen folks knew about hom lim and hom colims: start with a cat with a model structure, and "localize" it, ie formally invert all the weak equivalences. The new cat, called the homotopy category, is is general not well behaved as far as standard lims and colims: one has thus to introduce a new kind of universal objects appropriate for the homotopic context.

So, from the Annus Mirabilis begins a new chapter, but it does not end there. As a result of Quillen's shift, now many many "things" that were not under the rubric of homotopy theory, structures that are not even topological, acquire an homotopic flavor.

Funny enough, one of these is category theory itself: Cat, the category of small cats, has a default model structure, where weak equivalences are simply cats equivalence.

There were several attempts to generalize and expand on the look at Homotopy given by model structures, but if we focus on true radically new insights, here it goes:

back in the golden era, a standard homotopy was simply a continuous deformation, so essentially an invertible path between maps. In Quillen you start with the weak equivalences, but let us go back to the continuous deformation: if the cat where I want to introduce my weak equivalences happens to be a 2-cat, and I look at the groupoid of 2-maps therein, I have my continuous deformations.

So, here is the key insight that migrates from the modern approach to the post-modern: do not look at a single cat alone, but see it as as only the ground floor of a higher and higher groupoid (paths, paths of paths, etc.) Rather than being the bedrock of Homotopy, model structures become models, or presentations, of the REAL OBJECT of Homotopy, the invariant infinity groupoid in all its splendor. That basic insight is already in Grothendieck, around 1983, maybe earlier, but has blossomed into an entire field thanks to Voevodsky, Lurie, Rezk, etc.

What is fascinating, is that the post-modern era is not simply foundational, but admits also a foundationalist approach: a large swath of mathematics in principle can be seen from this angle, of "getting rid of equality", and replacing it with equivalences and their higher versions. .

$\endgroup$
4
  • $\begingroup$ I quote from Dmitri's answer "Quillen published his Homotopical Algebra in 1967, introducing model categories and using them in his Rational homotopy theory around 1968. Around 1972, he introduced higher algebraic K-theory." Because of this, I didn't mention model categories as they were already on the list $\endgroup$ Jun 17 at 15:52
  • 1
    $\begingroup$ @DavidWhite apologies. I do see it now, so I will edit out from my answer. $\endgroup$ Jun 17 at 15:55
  • $\begingroup$ Don't you mean 1952 for Eilenberg and Steenrod? $\endgroup$ Jun 17 at 23:18
  • $\begingroup$ Yes,: Eilenberg, Samuel; Steenrod, Norman E. (1952). Foundations of algebraic topology. Princeton, New Jersey: Princeton University Press. MR 0050886. (fixed) $\endgroup$ Jun 18 at 0:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.