nLab
looping

Idea

For $X$ any kind of space (or possibly a directed space, viewed as some sort of category or higher dimensional analogue of one), its loop space objects ${\Omega }_{x}X$ canonically inherit a monoidal structure, coming from concatenation of loops.

If $x\in X$ is essentially unique, then ${\Omega }_{x}X$ equipped with this monoidal structure remembers all of the structure of $X$: we say $X\simeq B{\Omega }_{x}X$ call $BA$ the delooping of the monoidal object $A$.

What all these terms (“loops” $\Omega$, “delooping” $B$ etc.) mean in detail and how they are presented concretely depends on the given setup. We discuss some of these below in the section Examples.

Examples

For topological spaces and $\mathrm{\infty }$-groupoids

There is an equivalence of (∞,1)-categories

$$\mathrm{\infty }{\mathrm{Grpd}}_{*}\stackrel{\stackrel{B}{\leftarrow }}{\underset{\Omega }{\to }}\mathrm{\infty }\mathrm{Group}$$\infty Grpd_* \stackrel{\overset{B}{\leftarrow}}{\underset{\Omega}{\to}} \infty Group

between pointed connected ∞-groupoids and ∞-groups, where $\Omega$ forms loop space objects.

This is presented by a Quillen equivalence of model categories

$${\mathrm{sSet}}_{*}\stackrel{\stackrel{\overline{W}}{\leftarrow }}{\underset{G}{\to }}\mathrm{sGrp}$$sSet_* \stackrel{\overset{\bar W}{\leftarrow}}{\underset{G}{\to}} sGrp

between the model structure on reduced simplicial sets and the transferred model structure on simplicial groups along the forgetful functor to the model structure on simplicial sets.

(See groupoid object in an (infinity,1)-category for more details on this Quillen equivalence.)

For parameterized $\mathrm{\infty }$-groupoids ($\mathrm{\infty }$-stacks / $(\mathrm{\infty },1)$-sheaves)

The following result makes precise for parameterized ∞-groupoids – for ∞-stacks – the general statement that $k$-fold delooping provides a correspondence between n-categories that have trivial r-morphisms for $r<k$ and k-tuply monoidal n-categories.

Definition

An Ek-algebra object $A$ in an (∞,1)-topos $H$ is called groupal if its connected components ${\pi }_0(A)\in H_{\le 0}$ is a group object.

Write ${\mathrm{Mon}}_{𝔼[k]}^{\mathrm{gp}}(H)$ for the (∞,1)-category of groupal $E_k$-algebra objects in $H$.

A groupal $E_1$-algebra – hence an groupal A-∞ algebra object in $H$ – we call an ∞-group in $H$. Write $\mathrm{\infty }\mathrm{Grp}(H)$ for the (∞,1)-category of ∞-groups in $H$.

Theorem (k-tuply monoidal $\mathrm{\infty }$-stacks)

Let $k>0$, let $H$ be an (∞,1)-category of (∞,1)-sheaves and let $H_{*}^{\ge k}$ denote the full subcategory of the category $H_{*}$ of pointed objects, spanned by those pointed objects thar are $k-1$-connected (i.e. their first $k$ homotopy sheaves) vanish. Then there is a canonical equivalence of (∞,1)-categories

$$H_{*}^{\ge k}\simeq {\mathrm{Mon}}_{𝔼[k]}^{\mathrm{gp}}(H).$$\mathbf{H}_*^{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H}) \,.

between the $(k-1)$-connected pointed objects and the groupal Ek-algebra objects in $H$.

This is (Lurie, Higher Algebra, theorem 5.1.3.6).

Specifically for $H=$ Top, this reduces to the classical theorem by Peter May

Theorem (May recognition theorem)

Let $Y$ be a topological space equipped with an action of the little cubes operad ${𝒞}_k$ and suppose that $Y$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space ${\Omega }^{k}X$ for some pointed topological space $X$.

This is EkAlg, theorem 1.3.16.

Remark

For $k=1$ we have a looping/delooping equivalence

$$H_{*}\stackrel{\stackrel{B}{\leftarrow }}{\underset{\Omega }{\to }}\mathrm{\infty }\mathrm{Grpd}(H)$$\mathbf{H}_* \stackrel{\overset{\mathbf{B}}{\leftarrow}}{\underset{\Omega}{\to}} \infty Grpd(\mathbf{H})

between pointed connected objects in $H$ and grouplike A-∞ algebra objects in $H$: ∞-group objects in $H$.

Remark

If the ambient (∞,1)-topos has homotopy dimension 0 then every connected object $E$ admits a point $*\to E$. Still, the (∞,1)-category of pointed connected objects differs from that of unpointed connected objects (because in the latter the natural transformations may have nontrivial components on the point, while in the former case they may not).

The connected objects $E$ which fail to be ∞-groups by failing to admit a point are of interest: these are the ∞-gerbes in the (∞,1)-topos.

For cohesive $\mathrm{\infty }$-groupoids

A special case of the parameterized $\mathrm{\infty }$-groupoids above are cohesive ∞-groupoids. Looping and delooping for these is discussed at cohesive (∞,1)-topos -- structures in the section Cohesive ∞-groups.

For $(\mathrm{\infty },n)$-categories

See delooping hypothesis.

Relation to looping and suspension

For $A$ any monoidal space, we may forget its monoidal structure and just remember the underlying space. The formation of loop space objects composed with this forgetful functor has a left adjoint $\Sigma$ which forms suspension objects.

Related concepts

• loop space object, free loop space object,

  • delooping, looping and delooping

  • loop space, free loop space, derived loop space

• suspension object

  • suspension, reduced suspension

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ Γ-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

Section 6.1.2 of

• Jacob Lurie, Higher Topos Theory

Section 5.1.3 of

• Jacob Lurie, Higher Algebra