homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
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 $B A$ 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.
There is an equivalence of (∞,1)-categories
between pointed connected ∞-groupoids and ∞-groups, where $\Omega$ forms loop space objects.
This is presented by a Quillen equivalence of model categories
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.)
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 $r \lt k$ and k-tuply monoidal n-categories.
An Ek-algebra object $A$ in an (∞,1)-topos $\mathbf{H}$ is called groupal if its connected components $\pi_0(A) \in \mathbf{H}_{\leq 0}$ is a group object.
Write $Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H})$ for the (∞,1)-category of groupal $E_k$-algebra objects in $\mathbf{H}$.
A groupal $E_1$-algebra – hence an groupal A-∞ algebra object in $\mathbf{H}$ – we call an ∞-group in $\mathbf{H}$. Write $\infty Grp(\mathbf{H})$ for the (∞,1)-category of ∞-groups in $\mathbf{H}$.
Let $k \gt 0$, let $\mathbf{H}$ be an (∞,1)-category of (∞,1)-sheaves and let $\mathbf{H}_*^{\geq k}$ denote the full subcategory of the category $\mathbf{H}_{*}$ 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
between the pointed $(k-1)$-connected objects and the groupal Ek-algebra objects in $\mathbf{H}$.
This is (Lurie, Higher Algebra, theorem 5.1.3.6).
Specifically for $\mathbf{H} =$ Top, this reduces to the classical theorem by Peter May
Let $Y$ be a topological space equipped with an action of the little cubes operad $\mathcal{C}_k$ 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.
For $k = 1$ we have a looping/delooping equivalence
between pointed connected objects in $\mathbf{H}$ and grouplike A-∞ algebra objects in $\mathbf{H}$: ∞-group objects in $\mathbf{H}$.
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.
A special case of the parameterized $\infty$-groupoids above are cohesive ∞-groupoids. Looping and delooping for these is discussed at cohesive (∞,1)-topos -- structures in the section Cohesive ∞-groups.
See delooping hypothesis.
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.
loop space object, free loop space object,
delooping, looping and delooping
Section 6.1.2 of
Section 5.1.3 of