delooping hypothesis


Higher category theory

higher category theory

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The delooping hypothesis is one of the “guiding hypotheses of higher category theory.” Like the homotopy hypothesis, it is generally accepted to be a “litmus test” that any suitable definition of n-category should satisfy. It states that:

The identification involves a degree shift: the ii-morphisms of a kk-tuply monoidal nn-category become (i+j)(i+j)-morphisms in the associated (kj)(k-j)-tuply monoidal (n+j)(n+j)-category.

Here jj-(simply) connected means that any two parallel ii-morphisms are equivalent for iji \leq j. Also, 00-tuply monoidal is interpreted as meaning pointed. We may also allow nn to be of the form (n,r) or (,r)(\infty,r), with the usual conventions that (n,r)+j=(n+j,r+j)(n,r)+j=(n+j,r+j), +j=\infty+j=\infty, and so on. In particular, taking j=kj=k we have:

  • kk-tuply monoidal nn-categories can be identified with pointed (k1)(k-1)-connected (n+k)(n+k)-categories.

The (n+j)(n+j)-category associated to a kk-tuply monoidal nn-category CC is called its jj-fold delooping and sometimes written B jCB^j C. Conversely, any kk-tuply monoidal nn-category CC with a point *C*\in C has a loop space object ΩC=C(*,*)\Omega C = C(*,*) which is a (k+1)(k+1)-tuply monoidal (n1)(n-1)-category.


  • Not infrequently the delooping hypothesis is used to supply a definition of “kk-tuply monoidal nn-category.” See k-tuply monoidal n-category for an investigation in low dimensions.

  • The delooping hypothesis is closely related to the stabilization hypothesis. In delooping language, the stabilization hypothesis says that once you have an nn-category that can be delooped n+2n+2 times, it can automatically be delooped infinitely many times.

  • In low dimensions, the delooping hypothesis is a special case of the exactness hypothesis.

In homotopy theory

A “groupoidal” version of the delooping hypothesis may be stated as

  • kk-tuply groupal nn-groupoids can be identified with (kj)(k-j)-tuply groupal (j1)(j-1)-connected (n+j)(n+j)-groupoids, for 0jk0\le j\le k.

Here “groupal” means “monoidal and such that all objects have inverses.” (This can actually be seen as a special case of the delooping hypothesis for kk-tuply monoidal (n,r)(n,r)-categories with rr set to 1-1.)

When n=n=\infty the groupoidal delooping hypothesis can be interpreted (via the homotopy hypothesis) as a standard result of classical homotopy theory: “grouplike E kE_k-spaces” can be delooped kk times. In particular, grouplike A A_\infty-spaces can be delooped once, and grouplike E E_\infty-spaces? can be delooped infinitely many times (producing a spectrum).

Non-grouplike A A_\infty-spaces can also be “delooped” in classical homotopy theory, but can only be recovered from their delooping up to group completion. This is because classical homotopy theory only works with (,0)(\infty,0)-categories, while the higher-categorical delooping of a non-grouplike A A_\infty-space (that is, a monoidal (,0)(\infty,0)-category) should be an (,1)(\infty,1)-category, not an (,0)(\infty,0)-category.

For (,n)(\infty,n)-categories

The delooping hypothesis also makes sense for (,n)(\infty,n)-categories rather than nn-categories. In this case a proof (using iterated (,1)(\infty,1)-categorical enrichment to define (,n)(\infty,n)-categories) is in

Last revised on September 2, 2015 at 13:27:54. See the history of this page for a list of all contributions to it.