nLab delooping hypothesis

Idea

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:

• k-tuply monoidal n-categories can be identified with $(k-j)$-tuply monoidal, $(j-1)$-simply connected $(n+j)$-categories, for any $0\le j \le k$.

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:

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

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.

Remarks

• 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.

In homotopy theory

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

• $k$-tuply groupal $n$-groupoids can be identified with $(k-j)$-tuply groupal $(j-1)$-connected $(n+j)$-groupoids, for $0\le j\le k$.

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 $k$-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.

For $(\infty,n)$-categories

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

• David Gepner, Rune Haugseng, Enriched ∞-categories via non-symmetric ∞-operads, arXiv