Homotopy hypothesis

In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy-theoretically speaking, that the ∞-groupoids are spaces.

One version of the hypothesis was claimed to be proved in the 1991 paper by Kapranov and Voevodsky.¹² Their proof turned out to be flawed and their result in the form interpreted by Carlos Simpson² is now known as the Simpson conjecture.³

In higher category theory, one considers a space-valued presheaf instead of a set-valued presheaf in ordinary category theory. In view of homotopy hypothesis, a space here can be taken to an ∞-groupoid.

A precise formulation of the hypothesis very strongly depends on the definition of an ∞-groupoid. One definition is that, mimicking the ordinary category case, an ∞-groupoid is an ∞-category in which each morphism is invertible or equivalently its homotopy category is a groupoid.

Now, if an ∞-category is defined as a simplicial set satisfying the weak Kan condition, as done commonly today, then ∞-groupoids amounts exactly to Kan complexes (= simplicial sets with the Kan condition) by the following argument. If ${\displaystyle X}$ is a Kan complex (viewed as an ∞-category) and ${\displaystyle f}$ a morphism in it, consider ${\displaystyle \sigma :\Lambda _{0}^{2}\to X}$ from the horn such that ${\displaystyle \sigma (0\to 1)=f,\,\sigma (0\to 2)=\operatorname {id} }$. By the Kan condition, ${\displaystyle \sigma }$ extends to ${\displaystyle {\overline {\sigma }}:\Delta ^{2}\to X}$ and the image ${\displaystyle g={\overline {\sigma }}(1\to 2)}$ is a left inverse of ${\displaystyle f}$. Similarly, ${\displaystyle f}$ has a right inverse and so is invertible. The converse, that an ∞-groupoid is a Kan complex, is less trivial and is due to Joyal (see Joyal's theorem).⁴⁵⁶

Because of the above fact, it is common to define ∞-groupoids simply as Kan complexes. Now, a theorem of Milnor and CW approximation say that Kan complexes completely determine the homotopy theory of (reasonable) topological spaces. So, this essentially proves the hypothesis. In particular, if ∞-groupoids are defined as Kan complexes (bypassing Joyal’s result), then the hypothesis is almost trivial.

However, if an ∞-groupoid is defined in different ways, then the hypothesis is usually still open. In particular, the hypothesis with Grothendieck's original definition of an ∞-groupoid is still open.

There is also a version of homotopy hypothesis for (weak) n-groupoids, which roughly says⁷⁸

The statement requires several clarifications:

• An n-groupoid is typically defined as an n-category where each morphism is invertible. So, in particular, the meaning depends on the meaning of an n-category (e.g., usually some weak version of an n-category),
• "the same as" usually means some equivalence (see below), and the definition of an equivalence typically uses some higher notions like an ∞-category,
• A homotopy n-type means a reasonable topological space with vanishing i-th homotopy groups, i > n at each base point (so a homotopy n-type here is really a weak homotopy n-type to be precise).

Moreover, the equivalence between the two notions is supposed to be given on one direction by a higher version of a fundamental groupoid, or the fundamental n-groupoid ${\displaystyle \Pi _{n}(X)}$ of a space X where⁹¹⁰

• an object is a point in X,
• a 1-morphism ${\displaystyle f:x\to y}$ is a path from a point x to a point y, with the compositions the concatenation of two paths,
• a 2-morphism is a homotopy from a path ${\displaystyle f:x\to y}$ to a path ${\displaystyle g:x\to y}$,
• a 3-morphism is a "map" between homotopies,
• and so on until n-morphisms.

The other direction is given by geometric realization.

This version is still open.

See also: Eilenberg–MacLane space, crossed module.