This page is to provide non-technical or maybe semi-technical discussion of the nature and role of homotopy theory. For more technical details and further pointers see at homotopy theory.
In one of the first textbooks on homotopy theory, An introduction to homotopy theory by Peter J. Hilton (1953), one reads:
Since the introduction of homotopy groups by Hurewicz in 1935, homotopy theory has been occupying an increasingly prominent place in the field of algebraic topology. Important new advances are continually being made in the subject by various workers; and the recent developments emanating from the French school of topologists underline the desirability of having available a basic introduction to homotopy theory suitable for those who wish to undertake research in the subject and for those who wish to be in a position to understand the modern techniques and results.
Sze-Tsen Hu writes in Homotopy Theory (1959):
The recognition of the branch of mathematics now called homotopy theory took place in the few years after the introduction of homotopy groups by Witold Hurewicz in 1935. Since then, with numerous advances made by various workers, it has been playing an increasingly important role in the expanding field of algebraic topology. However, there exists no textbook on the subject at any level except the extremely condensed Cambridge tract of P. J. Hilton entitled “An Introduction to Homotopy Theory.”
Thus, at the time homotopy theory was clearly identified as a part of algebraic topology.
The 1995 Handbook of Algebraic Topology (edited by Ioan M. James) is overwhelmingly devoted to homotopy theory.
However, the attitudes have changed since then. Haynes Miller writes in the Handbook of Homotopy Theory (2019):
This volume may be regarded as a successor to the “Handbook of Algebraic Topology,” edited by Ioan James and published a quarter of a century ago. In calling it the “Handbook of Homotopy Theory,” I am recognizing that the discipline has expanded and deepened, and traditional questions of topology, as classically understood, are now only one of many distinct mathematical disciplines in which it has had a profound impact and which serve as sources of motivation for research directions within homotopy theory proper.
Clark Barwick writes in The future of homotopy theory (2018):
Neither our subject nor its interaction with other areas of inquiry is widely understood. Some of us^4 call ourselves algebraic topologists, but this has the unhelpful effect of making the subject appear to be an area of topology, which I think is profoundly inaccurate. It so happens that one way (and historically the first way) to model homotopical thinking is to employ a very particular class of topological spaces.^5 Today, the praxis of homotopy theory interacts with topology no more often than it does with arithmetic geometry and category theory, and the interactions with areas like representation theory are growing rapidly. Homotopy theory is not a branch of topology. This is important, because as long as homotopy theory is classified under the umbrella of topology, there will be errors of judgement in who is considered competent to judge our work; the results of this at journals, on the job market, and in funding is real and lasting.
4 not me
5 I think of homotopy theory as an enrichment of the notion of equality, dedicated to the primacy of structure over properties. Simplistic and abstract though this idea is, it leads rapidly to a whole universe of nontrivial structures.
We list some mathematical objects that are undoubtedly studied by homotopy theory:
The first object is known under various names: ∞-groupoid, (homotopy) space, (homotopy) type, more recent (and so far exotic) names include “anima”. See the section on “spaces” for an explanation of why so many different names are used for (essentially) the same concept. These objects can be modeled by simplicial sets up to simplicial weak equivalence, or topological spaces up to weak homotopy equivalence, or many other models. See the section on models for more details about models.
Spectra or stable homotopy types.
Specific models include sequential spectra or symmetric spectra (valued in simplicial sets or topological spaces), orthogonal spectra, reduced excisive functors, etc.
∞-categories, also known as (∞,1)-categories, homotopy theories, etc.
Specific models include relative categories, quasicategories, simplicial categories, Segal categories, complete Segal spaces, etc. See the section on ∞-categories for more information.
(∞,n)-categories, which can be modeled by globular n-fold complete Segal spaces, Rezk’s Theta-n spaces, etc.
As already explained in the introduction, there is a recent tendency to separate homotopy theory from algebraic topology. This leaves the question as to what exactly algebraic topology is once the separation happens.
We list some mathematical objects that are undoubtedly studied by algebraic topology:
manifolds, as studied in surgery theory, bordism theory, etc.
These include topological manifolds, PL-manifolds, and smooth manifolds.
knots, tangles, surfaces, and other objects arising from manifolds.
These are some of the first and most important objects introduced in homotopy theory.
This is witnessed by the many names given to them, which all describe the same concept but have a slightly different emphasis:
Yet even this large variety of viewpoints is sometimes deemed insufficient, which leads mathematicians to introduce further names for such objects, such as “anima” used by Dustin Clausen and Peter Scholze, alluding to the process of vertical categorification, in its specific incarnation as groupoidal categorification, alias homotopification, which can be seen as “animating” 1-categorical objects by introducing higher homotopies into them.
We emphasize that these objects (except for types in homotopy type theory) come with an associated notion of weak equivalence, which is not isomorphism. For example, when using topological spaces as models for (homotopy) spaces, we are not interested in invariants under homeomorphisms, but rather only under weak homotopy equivalences. See the section on models for more details.
Informal ideas of objects studied by homotopy theory, such as (homotopy) spaces or (∞,1)-categories, must be presented using rigorous mathematical definitions. As of April 2020, this most likely means defining them using some flavor of set theory, although homotopy type theory may emerge as an alternative in the future.
This process introduces artifacts that are not present in the original informal idea and have no meaning in that context. As an example, suppose we present a (homotopy) space using a simplicial set. Then the set of 0-simplices or $n$-simplices for any $n\ge0$ is a piece of data specific to our chosen presentation. Another presentation may have a set of 0-simplices of different cardinality, say. On the other hand, if we take the set of connected components of a simplicial set, the resulting set describes a property of the underlying (homotopy) space and not just some specific presentation. This can be formalized by saying that the set of connected components is invariant under simplicial weak equivalences.
Thus simplicial sets up to simplicial weak equivalences model (homotopy) spaces.
One common way to encode models is to organize them in a relative category: the underlying category encodes the specific presentation, whereas weak equivalences encode the relevant notion of “sameness”.
The short answer is: we do not know how to work without models.
A good example is the notion of an (∞,1)-category, which exists at the Platonic level, and whose “shadows” (using Plato’s terminology) in the world of rigorous mathematics have been exhibited as various models of (∞,1)-categories: relative categories, quasicategories, Segal categories, complete Segal spaces, simplicial categories, etc.
However, the concept of (∞,1)-categories per se resists formalization in a satisfactory way. Homotopy type theory may succeed one day on this path, but so far no working definition has been exhibited yet.
An (∞,1)-category is a category enriched in (homotopy) spaces.
This can be formalized in many different ways. For example, if we model homotopy spaces by simplicial sets with simplicial weak equivalences, then we can consider categories enriched in simplicial sets, equipped with Dwyer–Kan equivalences, which are enriched functors that induce a simplicial weak equivalence on each mapping simplicial set, and after passing to the homotopy category (which in this case amounts to replacing each mapping simplicial set with its set of connected components), we get an equivalence of categories.
As one can see from this description, there are substantial modeling issues related to (∞,1)-categories.
Other models for (∞,1)-categories include relative categories, quasicategories, Segal categories, complete Segal spaces, etc.
All these models are equivalent in a precise sense: we can organize each of them into a Quillen model category so that all resulting model categories are Quillen equivalent.
The term “∞-category” has two different meanings: it can either mean (∞,1)-category, or (more recently) it can mean a quasicategory (as used in Lurie‘s books, for example).
No, for example, the cardinality of the set of 0-simplices of a small quasicategory does not have any model-independent meaning, although it is an isomorphism invariant.
Unfortunately, one often sees the adjective “model-independent” (ab)used to mean that only constructions that are meaningful on the level of underlying (∞,1)-categories are used in a particular argument with quasicategories. Of course, there is nothing special about quasicategories in this context: we often write “model-independent” arguments of such type also using model categories, for example.
See homotopy type theory FAQ for a detailed explanation. Here we only offer a highly impressionistic description.
Consider the traditional mathematical setup for (homotopy) spaces modeled by simplicial sets and simplicial weak equivalences: we start with first-order logic, introduce Zermelo-Fraenkel axioms, then define simplicial sets and simplicial weak equivalences.
Homotopy type theory replaces this entire stack with a flavor of type theory that aims to axiomatize
homotopy types (known in the field simply as types) directly, without these intermediate steps. In particular, first-order logic is subsumed into type theory. The latter (very) roughly resembles ETCS in how it chooses to formulate things like existence of (co)products of sets etc., i.e., these concepts are formulated in a categorical way rather than through constructions like Kuratowski pairs etc.
More fundamentally, the meaning of equality sign $=$ is altered: now $A=B$ (very) roughly means the the space of paths from the point $A$ to the point $B$ in whatever ambient space contains the points $A$ and $B$. Very roughly, if this space is empty, we could interpret this as $A\ne B$ and if it is nonempty, we could interpret this as $A=B$. However, homotopy type theory preserves the whole homotopy type of the path space and not just whether it is empty or not. The (Platonic) meaning of the equality sign is thus fundamentally enhanced and homotopy type theory explains how to manipulate such enhanced equalities.
This setup allows us to talk about homotopy types directly, without using models. Furthermore, the analog of weak equivalences in this setting behaves similar to isomorphisms (with a caveat that equality is now understood in the enhanced sense). In particular, every equivalence is invertible, unlike (say) simplicial weak equivalences, which need not be.
Much has been achieved on this path, including a new proof of the Blakers-Massey theorem that led to new classical consequences after being instantiated in (∞,1)-toposes.
However, one serious obstacle to adopting this approach in mainstream homotopy theory is that there is currently (April 2020) no working definition of an (∞,1)-category, although research is being conducted in this direction.
Homotopical Algebra is the title of a 1967 book by Daniel Quillen that introduced model categories.
Ever since then this term has been used to describe research in the area of model categories.
It is often hard to separate homotopical algebra from homotopy theory. Typically, “homotopical algebra” is used to refer to arguments that use notions from model categories rather explicitly, as opposed to (say) quasicategorical arguments. We do note that setting up the theory of quasicategories relies heavily on homotopical algebra and much of Lurie’s Higher Topos Theory is devoted to rather subtle aspects of homotopical algebra.
If we discard cofibrations and fibrations from the data of a model category, we get a relative category. Relative categories? are one of many equivalent models for (∞,1)-categories, so any model category has an underlying (∞,1)-category.
Since the underlying (∞,1)-category only depends on weak equivalences and not on cofibrations or fibrations, we deduce that two model categories with the same weak equivalences, but different cofibrations or fibrations have the same underlying (∞,1)-category.
Thus, the data of cofibrations and fibrations merely enhances the given presentation of an (∞,1)-category as a relative category with additional data that helps to organize various computations, e.g., to derive functors.
However, the ultimate answer to any computation with model categories that makes sense on the level of underlying (∞,1)-categories does not depend on the choice of cofibrations and fibrations, although its specific model certainly does.
Another important aspect of the relation between model categories and (∞,1)-categories is that a model category must satisfy additional (co)completeness conditions. In the original definition by Quillen, a model category must be finitely (co)complete, and its (co)fibrations need not be closed under retracts. In this case, the underlying (∞,1)-category admits finite ∞-(co)limits. In the revised definition by Kan, a model category must be (co)complete, and its (co)fibrations must be closed under retracts. In this case, the underlying (∞,1)-category admits small ∞-(co)limits. Finally, Hovey’s definition additionally requires factorizations to be functorial. This condition does not seem to alter the class of underlying (∞,1)-categories.
Homotopy theory serves as a foundation for much of modern (higher) category theory. For instance, the proof of equivalence of various models of (∞,n)-categories heavily relies on homotopical algebra. It is difficult to draw a sharp boundary between the two subjects; (∞,1)-categories could definitely be seen as being part of both, for example.
Homotopy theory also somewhat implicitly permeates classical category theory. For example, the correct notion of a pullback for diagrams of categories is not the strict pullback, but rather incorporates an additional isomorphism between the images of two given objects.
But this is precisely the homotopy pullback in the category of small categories equipped with the natural model structure.
Just as relative categories model (∞,1)-categories, we can expect functors between relative categories to model (∞,1)-functors.
The easiest way to perform such modeling is to require functors to preserve weak equivalences. This indeed works perfectly well. However, many functors arising in practice, such as the tensor product of chain complexes (with quasi-isomorphisms as weak equivalences) does not satisfy such a strong condition.
“Deriving” is a process that models (∞,1)-functors via functors between relative categories that need not preserve weak equivalences.
Multiple nonequivalent definitions of derived functors exist in the literature, and functors that can be derived in one of the definitions need not be derivable in another.
Typically (in most definitions), deriving a functor $F\colon C\to D$ between relative categories $C$ and $D$ involves replacing $F$ with $F\circ R$, where $R\colon C\to C$ is a resolution functor, which is typically require to preserve weak equivalences and be itself a weak equivalence of relative categories (known as a Dwyer-Kan equivalence of relative categories).
A definition with rather good properties was given recently by Hinich and relies on converting a given functor $F\colon C\to D$ into a cocartesian fibration (or cartesian fibration) over the category $\{0\to1\}$, and equipping the total category of this fibration with an obvious notion of weak equivalence. If the resulting functor of relative categories is a (co)cartesian fibration of relative categories, we can convert it to a functor $C\to D$ that preserves weak equivalences, which is the (left or right) derived functor of $F$.
Other (older) definitions involve Kan extensions along localization functors to homotopy categories. These definitions do not have such nice theoretical properties as the definition considered above. For example, they tend to misbehave when we try to derive compositions of functors.
The homotopy category $Ho(C)$ of an (∞,1)-category $C$ is obtained by replacing each of the hom ∞-groupoids $hom(x,y)$ with its set of connected components: $hom_{Ho(C)}(x,y) := \pi_0(hom_C(x,y))$.
This informal description can be formalized in any model of (∞,1)-categories. For instance, for relative categories we can formally invert all weak equivalences, the resulting localization being the homotopy category. (We note that inverting morphisms up to homotopy instead of strictly simply recovers the underlying (∞,1)-category of a relative category.)
The homotopy category is a useful tool for extracting invariants from a given (∞,1)-category. For example, in the relative category of chain complexes with quasi-isomorphisms we can compute
However, passing to the homotopy category destroys a tremendous amount of information about the underlying (∞,1)-category. In particular, it is rarely possible to recover any information about (∞,1)-limits and (∞,1)-colimits from the homotopy category, other than the case of products and coproducts. Thus, typically the passage to the homotopy category happens near the end of an argument, where one no longer needs categorical constructions like limits and colimits.
A stable (∞,1)-category is an (∞,1)-category that admits finite (∞,1)-(co)limits and such that cartesian squares coincide with cocartesian squares and there is a morphism from the terminal object to the initial object.
As is clear from the definition, being a stable (∞,1)-category is a property of an (∞,1)-category. Roughly speaking, a triangulated category is the homotopy category of a stable (∞,1)-category equipped with the additional data of a homotopy cofiber of any morphism.
The above is not quite true. In reality, one axiomatizes triangulated categories as additive categories with additional data of distinguished triangles (basically, homotopy cofiber sequences) and suspension functor that satisfies a bunch of axioms. A counterexample due to Muro, Schwede, and Strickland constructs a triangulated category that does not arise as the homotopy category of an (∞,1)-category.
Triangulated categories are a variation on the theme of homotopy categories and suffer from the same set of defects.
In particular, computations with homotopy limits or homotopy colimits are extremely difficult or impossible beyond the simplest cases.
Much work was put into rectifying this defect of triangulated categories. The resulting constructions are generically known as “enhancements” and include notions such as pretriangulated dg-categories and stable derivators.
These definitions are rather complicated when compared to stable (∞,1)-categories, and derivators still suffer from the same problems as homotopy categories, albeit on a higher level.
Thus, stable (∞,1)-categories due to their simplicity and generality can be reasonably seen as an “ultimate enhancement” of triangulated categories that does not suffer from their defects.
The most developed models for stable (∞,1)-categories are stable model categories and stable quasicategories. The former are treated by Hovey in Chapter 7 of his book Model categories? and the latter are treated in Chapter 1 of Lurie‘s Higher Algebra.
The derived (∞,1)-category of an abelian category $A$ is an (∞,1)-category presented by the relative category whose underlying category is the category of chain complexes in $A$ and weak equivalences are quasi-isomorphisms. The homotopy category of the derived (∞,1)-category recovers the traditional derived category, which is a triangulated category.
The derived category of $A$ must not be confused with the homotopy category of chain complexes valued in $A$. The latter category is also defined as the homotopy category of the relative category of $A$-valued chain complexes, but with weak equivalences being chain homotopy equivalences instead of quasi-isomorphisms.
The derived category (as well as the homotopy category of chain complexes) suffers from the same limitations as homotopy categories and triangulated categories. In particular, any theoretical constructions that involve homotopy limits and homotopy colimits beyond the case of homotopy products or homotopy coproducts invariably run into severe difficulties, often insurmountable.
One way to resolve this issue is to work with one of the models of the derived (∞,1)-category instead. By far the most popular choice in the literature amounts to working with the relative category described above, often augmented by additional formalisms such as model categories.
Under additional conditions on $A$, the relative category of chain complexes with values in $A$ and quasi-isomorphisms can be equipped with various model structures, of which the most important ones are the projective model structure on chain complexes and injective model structure on chain complexes. These model structures provide a convenient conceptual framework for projective resolutions and injective resolutions.
Another treatment of derived (∞,1)-categories can be given using stable quasicategories. See Chapter 1 in Lurie‘s Higher Algebra.
Last revised on April 29, 2021 at 20:25:37. See the history of this page for a list of all contributions to it.