In generality, homotopy theory is the study of mathematical contexts in which functions or rather (homo-)morphisms are equipped with a concept of homotopy between them, hence with a concept of “equivalent deformations” of morphisms, and then iteratively with homotopies of homotopies between those, and so forth.
A key aspect is that in such homotopy theoretic contexts the concept of isomorphism is relaxed to that of homotopy equivalence: Where a morphism is regarded as invertible if there is a reverse function such that both composites are equal to the identity morphism, for a homotopy equivalence one only requires the composites to be homotopic to the identity. Regarding objects in a homotopical context up to homotopy equivalence this way is to regard them as homotopy types.
The classical example is the classical homotopy theory of topological spaces, where one considers topological spaces with continuous functions between them, and with the original concept of topological homotopies between these continuous functions. The category whose objects are topological spaces and whose morphisms are homotopy equivalence-class of continuous functions is also called the classical homotopy category.
The classical homotopy theory of topological spaces has many applications, for example to covering space theory, to classifying space theory, to generalized (Eilenberg-Steenrod) cohomology theory and many more. (See also at shape theory.) Accordingly, homotopy theory has a large overlap with algebraic topology.
The classical homotopy theory of topological spaces may be abstracted to yield an “abstract homotopy theory” that applies to a large variety of contexts. There are several more or less equivalent formalizations of the concept of “abstract homotopy theory”, including
The terminology model category is short for “category of models of homotopy types”. The idea here is to consider categories equipped with suitable extra structure and properties that encodes the existence of homotopies between all morphisms and convient means to handle and control these, in particular a means to construct the corresonding homotopy category.
For a detailed introduction to homotopy theory from this perspective see at Introduction to Homotopy Theory.
The approach of (∞,1)-categories to homotopy theory is meant to be more truthful to the intrinsic nature of homotopy theory. Instead of equipping an ordinary category with a extra concept of homotopy between its morphisms, here one regards the resulting structure as a higher category where the homotopies themselves appear as a kind of higher order morphisms, called 2-morphisms and where higher homotopies of homotopies are regarded as k-morphisms for all $k$.
The terminology “(∞,1)-category” signifies that homotopy theory is but one special case of general higher category theory, namely (∞,1)-categories (hence homotopy theories) are those infinity-categories in which all k-morphisms for $k \gt 1$ are invertible up to homotopy. If one drops this constraint, so that homotopies become “directed” then one might still speak of “directed homotopy theory”.
The archetypical example of an (∞,1)-category is the $(\infty,1)$-category ∞Grpd of ∞-groupoids, just as Set is the archetypical 1-category.
This turns out to be equivalent, as homotopy theories, to the classical homotopy theory of topological spaces if restricted to those that admit the structure of CW-complexes. Another homotopy theory equivalent to this archetypical one is the classical homotopy theory of simplicial sets, see also at simplicial homotopy theory.
But there are many other homotopy theories besides (the various incarnations of) this classical one. Important sub-classes of homotopy theories include
stable homotopy theory modeled by stable model categories and stable (∞,1)-categories, where the operations of looping and delooping are an equivalence. This includes the stable (∞,1)-category of spectra as well as those of sheaves of spectra.
geometric homotopy theory modeld by model toposes and (∞,1)-toposes, which are the homotopy theoretic analogs of ordinary toposes. This includes the homotopy theoretic analog of categories of sheaves, called (∞,1)-categories of (∞,1)-sheaves or of ∞-stacks, but it potentially also contains “elementary (∞,1)-toposes”.
The geometric homotopy theory of (∞,1)-toposes in particular serves as the foundation for higher geometry/derived geometry. This is relevant notably in the physics of gauge theory, where gauge transformations are identified with homotopies in geometric homotopy theory. For more on this see at geometry of physics -- homotopy types.
On the other hand, the incarnation of homotopy theory as homotopy type theory exhibits the remarkably foundational nature of homotopy theory. Contrary to its original appearance as a fairly complicated-looking theory built on top of classical set theory and classcal topology, homotopy theory turns out to be intrinsically simple: it arises from plain dependent type theory just by adopting a fully constructive attitude towards the concept of identity/equality, see at identity type for more on this. For exposition of this perspective see (Shulman 17).
A convenient, powerful, and traditional way to deal with (∞,1)-categories is to “present” them by 1-categories with specified classes of morphisms called weak equivalences : a category with weak equivalences or homotopical category. The idea is as follows. Given a category $C$ with a class $W$ of weak equivalences, we can form its homotopy category or category of fractions $C[W^{-1}]$ by adjoining formal inverses to all the morphisms in $W$. The (∞,1)-category presented by $(C,W)$“ can be thought of as the result of regarding $C$ as an $\infty$-category with only identity $k$-cells for $k\gt 1$, then adjoining formal inverses to morphisms in $W$ in the $\infty$-categorical sense; that is, making them into equivalences rather than isomorphisms. It is remarkable that most $(\infty,1)$-categories that arise in mathematics can be presented in this way.
As with presentations of groups and other algebraic structures, very different presentations can give rise to equivalent $(\infty,1)$-categories. For example, several different presentations of the $(\infty,1)$-category of $\infty$-groupoids are:
The latter three can hence be regarded as providing “combinatorial models” for the homotopy theory of topological spaces.
The value of working with presentations of $(\infty,1)$-categories rather than the $(\infty,1)$-categories themselves is that the presentations are ordinary 1-categories, and thus much simpler to work with. For instance, ordinary limits and colimits are easy to construct in the category of topological spaces, or of simplicial sets, and we can then use these to get a handle on $(\infty,1)$-categorical limits and colimits in the $(\infty,1)$-category of $\infty$-groupoids. However, we always have to make sure that we use only 1-categorical constructions that are homotopically meaningful, which essentially means that they induce $(\infty,1)$-categorical meaningful constructions in the presented $(\infty,1)$-category. In particular, they must be invariant under weak equivalence.
Most presentations of $(\infty,1)$-categories come with additional classes of morphisms, called fibrations and cofibrations, that are very useful in performing constructions in a homotopically meaningful way. Quillen defined a model category to be a 1-category together with classes of morphisms called weak equivalences, cofibrations, and fibrations that fit together in a very precise way (the term is meant to suggest “a category of models for a homotopy theory”). Many, perhaps most, presentations of $(\infty,1)$-categories are model categories. Moreover, even when we do not have a model category, we often have classes of cofibrations and fibrations with many of the properties possessed by cofibrations and fibrations in a model category, and even when we do have a model category, there may be classes of cofibrations and fibrations, different from those in the model structure, that are useful for some purposes.
Unlike the weak equivalences, which determine the “homotopy theory” and the $(\infty,1)$-category that it presents, fibrations and cofibrations should be regarded as technical tools which make working directly with the presentation easier (or possible). Whether a morphism is a fibration or cofibration has no meaning after we pass to the presented $(\infty,1)$-category. In fact, every morphism is weakly equivalent to a fibration and to a cofibration. In particular, despite the common use of double-headed arrows for fibrations and hooked arrows for cofibrations, they do not correspond to $(\infty,1)$-categorical epimorphisms and monomorphisms.
In a model category, a morphism which is both a fibration and a weak equivalence is called an acyclic fibration or a trivial fibration. Dually we have acyclic or trivial cofibrations. An object $X$ is called cofibrant if the map $0\to X$ from the initial object to $X$ is a cofibration, and fibrant if the map $X\to 1$ to the terminal# object is a fibration. The axioms of a model category ensure that for every object $X$ there is an acyclic fibration $Q X \to X$ where $Q X$ is cofibrant and an acyclic cofibration $X\to R X$ where $R X$ is fibrant.
topological homotopy theory, classical model structure on topological spaces,
simplicial homotopy theory, classical model structure on simplicial sets
For a (higher) category theorist, the following examples of model categories are perhaps the most useful to keep in mind:
The morphisms from $A$ to $B$ in the $(\infty,1)$-category presented by $(C,W)$ are zigzags $\stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow} \to \cdots$; these are sometimes called generalized morphisms. Many presentations (including every model category) have the property that any such morphism is equivalent to one with a single zag, as in $\stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow}$. In a model category, a canonical form for such a zigzag is $X \stackrel{\simeq}{\leftarrow} Q X \to R Y \stackrel{\simeq}{\leftarrow} Y$ where $Q X$ is cofibrant and $R Y$ is fibrant. In this case we can moreover take $Q X\to X$ to be an acyclic fibration and $Y\to R Y$ to be an acyclic cofibration.
Often it suffices to consider even shorter zigzags of the form $\stackrel{\simeq}{\leftarrow} \to$ or $\to \stackrel{\simeq}{\leftarrow}$. In particular, this is the case if every object is fibrant or every object is cofibrant. For example:
If $X$ is cofibrant and $Y$ is fibrant, then every generalized morphism from $X$ to $Y$ is equivalent to an ordinary morphism. For example, if $X$ is a cofibrant 2-category, then every pseudofunctor $X\to Y$ is equivalent to a strict 2-functor $X\to Y$
Quillen also introduced a highly structured notion of equivalence between model categories, now called a Quillen equivalence, which among other things ensures that they present the same $(\infty,1)$-category. Quillen equivalences are now being used to compare different definitions of higher categories.
equivariant homotopy theory, global equivariant homotopy theory
algebraic homotopy: In his talk at the 1950 ICM in Harvard, Henry Whitehead introduced the idea of algebraic homotopy theory and said
“The ultimate aim of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that ‘analytic’ is equivalent to ‘pure’ projective geometry.”
This theme was taken up by Baues, (1988), using a type of abstract homotopy theory closely related to Ken Brown’s categories of fibrant objects. Whitehead’s own work was extended by Ronnie Brown and Phil Higgins, see nonabelian algebraic topology.
deformation retract of a homotopical category, neighborhood retract
Hurewicz fibration, Hurewicz connection, Hurewicz cofibration
model structure on simplicial sets, model structure on dendroidal sets
An exposition with an eye towards homotopy type theory is in
The original axiomatization of homotopy theory by model categories is due to
The similar axiomatization involving the weaker structure of a calculus of fractions is due to
A standard account of the modern form of simplicial homotopy theory is in
Formulation of abstract homotopy theory as the theory of (∞,1)-toposes is due to
and the formalization of this in the internal language of homotopy type theory is due to
A foundational set of lecture notes is developing in
See also
H. J. Baues, Homotopy types, in Handbook of Algebraic Topology, (edited by I.M. James), North Holland, 1995.
Jasper Moller, Homotopy theory for beginners (pdf)
Julia Bergner, A survey of $(\infty,1)$-categories (arXiv).
William Dwyer, Homotopy theory and classifying spaces, Copenhagen, June 2008 (pdf)
William Dwyer, P .Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs, American Mathematical Society (2004) (there exists this pdf copy of what seems to be a preliminary version of this book)
K. H. Kamps and Tim Porter, Abstract Homotopy and Simple Homotopy Theory (GoogleBooks)
Emily Riehl, Categorical homotopy theory, Lecture notes (pdf)
Brief indications of open questions and future directions (as of 2013) of algebraic topology and stable homotopy theory are in
and in