homotopy theory, (∞,1)-category theory, homotopy type theory
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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.
Write
Grpd for the (2,1)-category of groupoids (objects are groupoids, 1-morphisms are functors between these and 2-morphisms are natural transformations between those, which are nessecarily natural isomorphisms),
Grp for the 1-category of groups (discrete groups), also regarded as a (2,1)-category;
$Grpd^{\ast/}$ for the $(2,1)$-category of pointed objects in Grpd,
$Grpd_{\geq 1} \hookrightarrow Grpd$ for the full sub-(2,1)-category of connected groupoids, those for which $\pi_0 \simeq \ast$;
$Grp^{\ast/}_{\geq 1}$ for the pointed objects in connected groupoids.
$\pi_1(X,x) \in Grp$ for the fundamental group of a pointed groupoid $(\ast \stackrel{x}{\to} X) \in Grpd^{\ast/}$ at the given basepoint.
$\mathbf{B}G \in Grpd$, given a group $G$, for the groupoid $(G\stackrel{\longrightarrow}{\longrightarrow} \ast)$, with composition given by the product in the group. There are two possible choices of conventions, we agree that
The (2,1)-category $Grp_{\geq 1}$ of connected groupoids is equivalent to its full sub-(2,1)-category on those objects of the form $\mathbf{B}G$, for $G$ a group.
Given a connected groupoid $X$, pick any basepoint $x\in X$ and consider the canonical inclusion $\mathbf{B}\pi_1(X,x) \longrightarrow X$. By construction this is fully faithful and by assumption of connectedness it is essentially surjective, hence it is an equivalence of groupoids.
The hom-groupoids between connected groupoids with fundamental groups $G$ and $H$, respectively, are equivalent to the action groupoids of the set of group homomorphisms $G \to H$ acted on by conjugation with elements of $H$:
Given two group homomorphisms $\phi_1, \phi_2 \colon G \longrightarrow H$ then an isomorphism between them in this hom-groupoid is an element $h \in H$ such that
By direct inspection of the naturality square for the natural transformations which are the morphisms in $Grpd(\mathbf{B}G, \mathbf{B}H)$:
The operation of forming $\pi_1$ is equivalently the operation of forming the homotopy fiber product of the point inclusion with itself, and hence extends to a (2,1)-functor
Restricted to connected groupoids among the pointed groupoids, the functor $\pi_1 \colon Grpd^{\ast/}_{\geq 1} \longrightarrow Grp$ of remark 1 is an equivalence of (2,1)-categories.
It is clear that the functor is essentially surjective: for $G$ any group then $\pi_1(\mathbf{B}G,\ast) \simeq G$.
The more interesting point to notice is that $\pi_1$ is indeed a fully faithful (2,1)-functor, in that for any $(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1}$ then the functor
is an equivalence of hom-groupoids. By prop. 1 it is sufficient to check this for $X = \mathbf{B}G$ and $Y = \mathbf{B}H$ with their canonical basepoints, hence to check that for any two groups $G,H$ the functor
is an equivalence.
To see this, observe that, by definition of pointed objects via the undercategory under the point, a morphism in $Grpd^{\ast/}$ between groupoids of this form $\mathbf{B}(-)$ is a diagram in $Grp$ (unpointed) of the form
where the natural isomorphism is equivalently just the choice of an element $h \in H$. Hence these morphisms are pairs $(\phi,h)$ of a group homomorphism and an element of the domain.
We claim that the (2,1)-functor $\pi_1$ takes such $(\phi,h)$ to the homomorphism $Ad_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H$. To see this, consider via remark 1 this functor as forming loops:
This shows that on a morphism as above this acts by forming the pasting
Unwinding the whiskering of natural transformations here, the claim follows, as indicated by the label of the upper 2-morphisms on the right.
One observes now that these extra labels $h$ are precisely the information that “trivializes” the conjugation action which in prop. 2 prevents the bare set of group homomorphism: a 2-morphism $(\phi_1, h_1) \Rightarrow (\phi_2,h_2)$ in $Grp^{\ast/}$ is a natural isomorphism of groupoids
(encoding a conjugation relation $\phi_2 = Ad_{h} \circ \phi_1$ as above) such that we have the pasting relation
But this says in components that $h_2 = h_1\cdot h$. Hence there is a at most one morphism in $Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))$ from $(\phi_1,h_1)$ to $(\phi_2,h_2)$: it exists if $\phi_2 = Ad_h \circ \phi_1$ and $h_2 = h_1\cdot h$.
But since, by the previous argument, the functor $\pi_1$ takes $(\phi_1,h_1)$ to $Ad_{h_1^{-1}} \circ \phi_1$, this means that such a morphism exists precisely if both $(\phi_1,h_1)$ and $(\phi_2,h_2)$ are taken to the same group homomorphism by $\pi_1$
This establishes that $\pi_1$ is alspo an equivalence on all hom-groupoids.
This proof also shows that $\mathbf{B}(-)$ is in fact the inverse equivalence:
There is an equivalence of (2,1)-categories between pointed connected groupoids and plain groups
given by forming loop space objects and by forming deloopings.
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