homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
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 $i$-morphisms of a $k$-tuply monoidal $n$-category become $(i+j)$-morphisms in the associated $(k-j)$-tuply monoidal $(n+j)$-category.
Here $j$-(simply) connected means that any two parallel $i$-morphisms are equivalent for $i \leq j$. Also, $0$-tuply monoidal is interpreted as meaning pointed. We may also allow $n$ to be of the form (n,r) or $(\infty,r)$, with the usual conventions that $(n,r)+j=(n+j,r+j)$, $\infty+j=\infty$, and so on. In particular, taking $j=k$ we have:
The $(n+j)$-category associated to a $k$-tuply monoidal $n$-category $C$ is called its $j$-fold delooping and sometimes written $B^j C$. Conversely, any $k$-tuply monoidal $n$-category $C$ with a point $*\in C$ has a loop space object $\Omega C = C(*,*)$ which is a $(k+1)$-tuply monoidal $(n-1)$-category.
Not infrequently the delooping hypothesis is used to supply a definition of ”$k$-tuply monoidal $n$-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 $n$-category that can be delooped $n+2$ 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 $k$-tuply monoidal $(n,r)$-categories with $r$ set to $-1$.)
When $n=\infty$ the groupoidal delooping hypothesis can be interpreted (via the homotopy hypothesis) as a standard result of classical homotopy theory: “grouplike $E_k$-spaces” can be delooped $k$ times. In particular, grouplike $A_\infty$-spaces can be delooped once, and grouplike $E_\infty$-spaces? can be delooped infinitely many times (producing a spectrum).
Non-grouplike $A_\infty$-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 $(\infty,0)$-categories, while the higher-categorical delooping of a non-grouplike $A_\infty$-space (that is, a monoidal $(\infty,0)$-category) should be an $(\infty,1)$-category, not an $(\infty,0)$-category.