Higher category theory
higher category theory
Extra properties and structure
Paths and cylinders
For 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 canonically inherit a monoidal structure, coming from concatenation of loops.
If is essentially unique, then equipped with this monoidal structure remembers all of the structure of : we say call the delooping of the monoidal object .
What all these terms (“loops” , “delooping” 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.
For plain groups delooping to groupoids
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;
for the -category of pointed objects in Grpd,
for the full sub-(2,1)-category on connected groupoids, those for which ;
for the pointed objects in connected groupoids.
for the fundamental group of a pointed groupoid at the given basepoint.
, given a group , for the groupoid , with composition given by the product in the group. There are two possible choices of conventions, we agree that
The hom-groupoids between connected groupoids with fundamental groups and , respectively, are equivalent to the action groupoids of the set of group homomorphisms acted on by conjugation with elements of :
Given two group homomorphisms then an isomorphism between them in this hom-groupoid is an element such that
By direct inspection of the naturality square for the natural transformations which are the morphisms in :
It is clear that the functor is essentially surjective: for any group then .
The more interesting point to notice is that is indeed a fully faithful (2,1)-functor, in that for any then the functor
is an equivalence of hom-groupoids. By prop. 1 it is sufficient to check this for and with their canonical basepoints, hence to check that for any two groups the functor
is an equivalence.
To see this, observe that, by definition of pointed objects via the undercategory under the point, a morphism in between groupoids of this form is a diagram in (unpointed) of the form
where the natural isomorphism is equivalently just the choice of an element . Hence these morphisms are pairs of a group homomorphism and an element of the domain.
We claim that the (2,1)-functor takes such to the homomorphism . 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 are precisely the information that “trivializes” the conjugation action which in prop. 2 prevents the bare set of group homomorphism: a 2-morphism in is a natural isomorphism of groupoids
(encoding a conjugation relation as above) such that we have the pasting relation
But this says in components that . Hence there is a at most one morphism in from to : it exists if and .
But since, by the previous argument, the functor takes to , this means that such a morphism exists precisely if both and are taken to the same group homomorphism by
This establishes that is alspo an equivalence on all hom-groupoids.
This proof also shows that 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.
For topological spaces and -groupoids
There is an equivalence of (∞,1)-categories
between pointed connected ∞-groupoids and ∞-groups, where 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.)
For parameterized -groupoids (-stacks / -sheaves)
The following result makes precise for parameterized ∞-groupoids – for ∞-stacks – the general statement that -fold delooping provides a correspondence between n-categories that have trivial r-morphisms for and k-tuply monoidal n-categories.
This is (Lurie, Higher Algebra, theorem 188.8.131.52).
Specifically for Top, this reduces to the classical theorem by Peter May
Theorem (May recognition theorem)
Let be a topological space equipped with an action of the little cubes operad and suppose that is grouplike. Then is homotopy equivalent to a -fold loop space for some pointed topological space .
This is EkAlg, theorem 1.3.16.
For cohesive -groupoids
A special case of the parameterized -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.
Relation to looping and suspension
For 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 which forms suspension objects.
Section 6.1.2 of
Section 5.1.3 of