Grothendieck's homotopy hypothesis, is, as the $n$lab states:
Theorem: There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$.
What are the applications of this hypothesis? Why is it so fundamental? Can it be "generalized", perhaps by using the following definition of spaces: a space is simply a sheaf of sets on some site $\mathbf{Loc}$ of local models with a Grothendieck topology $τ$ on it?
For me this result fits in a context of other results that give complete algebraic invariants for homotopy types. The broad program sometimes goes under the rubric Whitehead's algebraic homotopy program.
If we define a homotopy $n$-type (for $n \geq 1$) as an object of the localization of a suitable category of spaces (e.g., $Top$ or simplicial sets) with respect to maps $f: X \to Y$ that induce isomorphisms on homotopy groups $\pi_k(X, x) \to \pi_k(Y, f(x))$ for each choice of basepoint $x$ and $1 \leq k \leq n$, then it is well-known and classical that homotopy 1-types are classified by their fundamental groupoids, i.e., the localization is equivalent to the category of groupoids. The homotopy hypothesis can be seen as a far-reaching generalization of this basic result; the result is essentially a 1-dimensional truncation of the homotopy hypothesis.
Thus, we could extend this idea of $n$-truncating $\infty$-groupoids past $n = 1$. Homotopy $n$-types are thus classified by $n$-groupoids; it is interesting to see how this subsumes some of the classical results. For example, looking at connected 2-types, these are classified by groupal monoidal groupoids; passing to appropriate skeletal models, this means that connected 2-types $X$ are classified by triples $(\pi_1(X), \pi_2(X), k)$ where $k \in H^3(\pi_1(X), \pi_2(X))$ (an example of a $k$-invariant) is the class of a 3-cocycle
$$\pi_1(X) \times \pi_1(X) \times \pi_1(X) \to \pi_2(X)$$
that in essence specifies an associativity constraint for a monoidal category structure. This description is a modern rendering of work going back to Eilenberg and Mac Lane:
where the $k$-invariant of a 2-type $X$ can be described in terms of its Postnikov tower.
Along similar lines, Joyal and Tierney (in apparently unpublished work, but circa 1984) described algebraic 3-types in terms of Gray-enriched groupoids, which are appropriately strictified groupoidal tricategories.
Of course, we also know that homotopy types can be described in terms of Kan complexes (which are classical models for $\infty$-groupoids), which is one simplicial manifestation of the homotopy hypothesis. My understanding is that Grothendieck was interested in fundamental algebraic operations on a more globular type of structure that arises from homotopy $n$-types $X$, where the $j$-cells for $j < n$ are maps $D^j \to X$ (thinking here of $D^j$ as a co-globular space) and $n$-cells are maps $D^n \to X$ modulo homotopy rel boundary. Surely this type of structure fits roughly into the sequence of the "classical" weak $n$-categories (category, bicategory, tricategory). If I can indulge in some shameless self-promotion, a main motivation for the notion of weak $n$-category that I presented in 1999 (see Tom Leinster's article) was actually to give a definition of Grothendieck fundamental $n$-groupoids along those sorts of "classical" lines (this is somewhat predating the "$(\infty, 1)$-revolution").
I am not sure of the state of the art here, but if I can be allowed to speculate wildly (and with it being understood that I am not a homotopy theorist): we might begin by observing that objects like Kan complexes (or their truncations) are not, technically speaking, algebraic in the sense of being strictly described in terms of operations subject to equational axioms. They can be algebraized, and something like an algebraic notion of (weak) $n$-groupoid or $\infty$-groupoid results, but actually milking usable algebraic invariants out of such structures, i.e., extracting usable algebraic invariants for $n$-types in a way that extends the results of Eilenberg-Mac Lane, Joyal-Street, etc., seems to me a possibly worthy research project. Possibly such study would proceed by locating usable semi-strictifications of $n$-categorical structures (à la the manner in which Gray-groupoids are semistrict tricategorical groupoids). In other directions: researchers such as Ronnie Brown have worked on $n$-types via more cubical notions (cubical $\omega$-groupoids), especially with a view toward higher van Kampen theorems, and Baues (whose work I don't know) has also worked on Whitehead's program (see his book Algebraic Homotopy, I guess).
While we're in this speculative mode: I don't know what really to say to the suggestion that we generalize the HH to objects of a Grothendieck topos $E$ (whose objects are thought of as "generalized spaces"). The only thing that comes to my mind right away is that we could try defining homotopy groups of objects in $E$ by appealing to a suitable "geometric realization" that passes through a left-exact left adjoint $E \to \textbf{Simp-Set}$. But there could be many choices of such lex left adjoints; they correspond to "$E$-models" in simplicial sets, where model is in the sense of whatever geometric theory the Grothendieck topos $E$ classifies. This could be interesting, or could be a dead end. Hard for me to say.
There are several ways to interpret the homotopy hypothesis.
Strictly speaking, Grothendieck's homotopy hypothesis is not a theorem yet: Grothendieck stated it in a very precise way in the very first pages of Pursuing stacks, by defining explicitly a notion of weak $\infty$-groupoid and by associating functorially to any space $X$ its weak higher groupoid $\Pi_\infty(X)$. Grothendieck's notion of weak higher groupoid is very closely related to Batanin's (who restated the homotopy hypothesis using his formalism). In this sense the homotopy hypothesis might be seen as a syntactic statement. The theory of weak higher groupoids is based on a language (the one of globular objects=$\infty$-graphs), and the homotopy theory of weak higher groupoids has the nice property that, up to equivalence, every object is free in an appropriate sense (=being cofibrant).
This is illustrated as follows: consider your favourite $\infty$-graph and then the weak higher groupoid associated to it. Then, if you want to add a relation between two $n$-cells, you simply add freely a $(n+1)$-cell connecting them. This means that it is quite easy to actually define a weak higher groupoid out of any reasonable sentence formulated in the globular language, and that the resulting object will be characterized by a universal property both in a strict sense and in an homotopical sense. If you interpret such a language in the world of stacks of weak higher groupoids, this means that you will have a powerful way to understand geometric objects (having such a theory at hand is the main motivation of Grothendieck for the definition of weak higher groupoids).
The meaning of the homotopy hypothesis is related to the fact that the homotopy theory of CW-complexes is freely generated by the point under small homotopy colimits (and thus, any homotopy theory is canonically enriched in homotopy types). Therefore, the homotopy hypothesis means that the language of weak higher groupoids is not only convenient, but also universal: this is a very appropriate language for homotopy theory in general (in some sense, this is the language, or even better, the logic of CW-structures). I insist on language here because, after the work of Awodey, Voevodsky et al. on homotopy type theory, the homotopy hypothesis has even more meaning: the very syntax of (cofibrant) weak higher groupoids is very related to the formulation of what people from homotopy type theory call higher inductive types, and thus, a full understanding of the homotopy theory of higher groupoids (e.g. the homotopy hypothesis, seen as way to describe higher topoi as stacks of weak higher groupoids) would give powerful tools to this part of logic (and thus to people who create programming languages for proof assistants, for instance).
Now, the homotopy hypothesis, considered in the spirit of the nLab (and which should not be attributed to Grothendieck, then), is a theorem, if we define weak higher groupoids as Kan complexes (a theorem due to Milnor in the 1950's), but more importantly a principle to develop the theory of $(\infty,1)$-categories as the one of categories enriched in weak higher groupoids, without giving an explicit description of what such things are. But such a principle must be completed by another one: the language of ordinary category theory can be interpreted to speak of $(\infty,1)$-categories.
This latter principle can already be seen in the work of Dwyer and Kan on simplicial categories, but it has been promoted to another level with Joyal's insights: quasi-categories (aka Boardman and Vogt's weak Kan complexes) define a category in which the very syntax of category theory defines constructions which are homotopy invariant (i.e. define objects in the Joyal model category structure which can be described in terms of mapping spaces, homotopy (co)limits, etc), so that this syntactic interpretation can be translated in any other model of $(\infty,1)$-categories (as far as the translation is done through a zig-zag of Quillen equivalences, say). This gives very powerful tools to produce and characterize homotopy categories using the language of ordinary category theory only, and underlies all the beautiful work of Lurie and others.
In conclusion, we see that there are two problems here: having an appropriate synthetic language for the theory of $(\infty,1)$-categories (and then the language of ordinary category theory, together with the `easy' version of the homotopy hypothesis provide such a thing, through the non-trivial work of Joyal, Lurie, etc); or having an appropriate analytic language to speak of objects of higher topoi (and then, Grothendieck's homotopy hypothesis would give us such a thing). Of course, the best thing would be to have both of them.
Since "Pursuing Stacks" was "written in English in response to a correspondence in English" I feel I ought to explain my own current position in relation to the Grothendieck Programme, and also to that of Whitehead, many of whose results and methods were a kind of guiding light. In particular, his theorem on free crossed modules proved in "Combinatorial Homotopy II" (CHII) led us (Phil Higgins and I) away from attempting to define higher homotopy groupoids for spaces, but instead for pairs of spaces in dimension 2, and then for filtered spaces. This suggested analogous results in discussions with Jean-Louis Loday.
A general picture of the methodology is covered by the following "generic" diagram of categories and functors:
with the following properties:
The purpose of the final part of 1. is to allow some calculations, and in particular some description of certain "free" objects. 3. is little vague, but is useful to bear in mind. Finally, $B= U \mathbb B$, where $U$ is the forgetful functor, is a kind of classifying space for the algebraic model.
We have two examples of this situation.
A. topological data = filtered spaces, broad algebraic data = (strict) cubical $\omega$-groupoids with connections, narrow algebraic data = crossed complexes. This is explained in detail in the book Nonabelian Algebraic Topology, which develops a lot of the work in CHII, and does not reach other parts.
B, topological data = $n$-cubes of pointed spaces, broad algebraic data = cat$^n$-groups, narrow algebraic data = crossed $n$-cubes of groups. A survey of this may found in the expository article "Computing homotopy types using crossed N-cubes of groups", with references to the original papers, particularly Brown/Loday and Ellis/Steiner.
The not-so-easy equivalences between the two kinds of algebraic data are a key to the methods. The broad model is used for intuition, conjectures and proofs; the narrow algebraic model is used for relation to classical theory and for calculations. One can hop between these at will!
Criterion 1 also excludes a number of algebraic models, such as: simplicial groups; weak models; $2$-crossed modules, and others, either because they not directly defined homotopically or because they are not known to preserve certain colimits.
Many people may be uncomfortable with the use of "topological data", rather than spaces, but this is how the methods work, and some thought should be given to evaluating that. There is more discussion in a talk I gave in Paris, June 5, 2014, available on my preprint page, as are talks at Galway and Aveiro. Grothendieck's analysis of "topological spaces" is referred to in another answer.
My understanding of Whitehead's work is that he worked with many specific examples, or classes of examples, which had additional structure to that of a topological space.
Also between the "broad" and the "narrow" there are and will be many intermediate cases, of possible use.
Finally, when I told Grothendieck in 1985 that (strict) $n$-fold groupoids model homotopy $n$-types (Loday's result, in essence), he exclaimed: "But that is absolutely beautiful!" For many reasons, his thoughts on this were not developed.
Actually Loday's result is for the pointed case. For more general results, see this paper, Blanc & Paoli.
