n-category = (n,n)-category
n-groupoid = (n,0)-category
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 -morphisms of a -tuply monoidal -category become -morphisms in the associated -tuply monoidal -category.
Here -(simply) connected means that any two parallel -morphisms are equivalent for . Also, -tuply monoidal is interpreted as meaning pointed. We may also allow to be of the form (n,r) or , with the usual conventions that , , and so on. In particular, taking we have:
The -category associated to a -tuply monoidal -category is called its -fold delooping and sometimes written . Conversely, any -tuply monoidal -category with a point has a loop space object which is a -tuply monoidal -category.
Not infrequently the delooping hypothesis is used to supply a definition of ”-tuply monoidal -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 -category that can be delooped times, it can automatically be delooped infinitely many times.
In low dimensions, the delooping hypothesis is a special case of the exactness hypothesis.
A “groupoidal” version of the delooping hypothesis may be stated as
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 -tuply monoidal -categories with set to .)
When the groupoidal delooping hypothesis can be interpreted (via the homotopy hypothesis) as a standard result of classical homotopy theory: “grouplike -spaces” can be delooped times. In particular, grouplike -spaces can be delooped once, and grouplike -spaces? can be delooped infinitely many times (producing a spectrum).
Non-grouplike -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 -categories, while the higher-categorical delooping of a non-grouplike -space (that is, a monoidal -category) should be an -category, not an -category.