nLab delooping hypothesis

Context

Higher category theory

higher category theory

Contents

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.

Revised on October 6, 2010 22:51:17 by Urs Schreiber (87.212.203.135)