Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations

Homotopy theory



For XX 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 Ω xX\Omega_x X canonically inherit a monoidal structure, coming from concatenation of loops.

If xXx \in X is essentially unique, then Ω xX\Omega_x X equipped with this monoidal structure remembers all of the structure of XX: we say XBΩ xXX \simeq B \Omega_x X call BAB A the delooping of the monoidal object AA.

What all these terms (“loops” Ω\Omega, “delooping” BB 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 topological spaces and \infty-groupoids

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

Grpd 1 */ΩBGroup \infty Grpd^{\ast/}_{\geq 1} \stackrel{\overset{B}{\leftarrow}}{\underset{\Omega}{\to}} \infty Group

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

This is presented by a Quillen equivalence of model categories

sSet *GW¯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 \infty-groupoids (\infty-stacks / (,1)(\infty,1)-sheaves)

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


An Ek-algebra object AA in an (∞,1)-topos H\mathbf{H} is called groupal if its connected components π 0(A)H 0\pi_0(A) \in \mathbf{H}_{\leq 0} is a group object.

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

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

Theorem (k-tuply monoidal \infty-stacks)

Let k>0k \gt 0, let H\mathbf{H} be an (∞,1)-category of (∞,1)-sheaves and let H * k\mathbf{H}_*^{\geq k} denote the full subcategory of the category H *\mathbf{H}_{*} of pointed objects, spanned by those pointed objects thar are k1k-1-connected (i.e. their first kk homotopy sheaves) vanish. Then there is a canonical equivalence of (∞,1)-categories

H k */Mon 𝔼[k] gp(H). \mathbf{H}^{\ast/}_{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H}) \,.

between the pointed (k1)(k-1)-connected objects and the groupal Ek-algebra objects in H\mathbf{H}.

This is (Lurie, Higher Algebra, theorem

Specifically for H=\mathbf{H} = Top, this reduces to the classical theorem by Peter May

Theorem (May recognition theorem)

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

This is EkAlg, theorem 1.3.16.


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

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

between pointed connected objects in H\mathbf{H} and grouplike A-∞ algebra objects in H\mathbf{H}: ∞-group objects in H\mathbf{H}.


If the ambient (∞,1)-topos has homotopy dimension 0 then every connected object EE admits a point *E* \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 EE which fail to be ∞-groups by failing to admit a point are of interest: these are the ∞-gerbes in the (∞,1)-topos.

For cohesive \infty-groupoids

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.

For (,n)(\infty,n)-categories

See delooping hypothesis.

Relation to looping and suspension

For AA 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 Σ\Sigma which forms suspension objects.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object


Section 6.1.2 of

Section 5.1.3 of

Revised on July 2, 2013 00:53:17 by Urs Schreiber (