Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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 plain groups delooping to groupoids



The (2,1)-category Grp 1Grp_{\geq 1} of connected groupoids is equivalent to its full sub-(2,1)-category on those objects of the form BG\mathbf{B}G, for GG a group.


Given a connected groupoid XX, pick any basepoint xXx\in X and consider the canonical inclusion Bπ 1(X,x)X\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 GG and HH, respectively, are equivalent to the action groupoids of the set of group homomorphisms GHG \to H acted on by conjugation with elements of HH:

Grpd(BG,BH)Grp(G,H)// adH Grpd(\mathbf{B}G, \mathbf{B}H) \simeq Grp(G,H)//_{ad}H

Given two group homomorphisms ϕ 1,ϕ 2:GH\phi_1, \phi_2 \colon G \longrightarrow H then an isomorphism between them in this hom-groupoid is an element hHh \in H such that

ϕ 2=Ad hϕ 1h 1ϕ 1()h. \phi_2 = Ad_h \circ \phi_1 \coloneqq h^{-1}\cdot \phi_1(-) \cdot h \,.

By direct inspection of the naturality square for the natural transformations which are the morphisms in Grpd(BG,BH)Grpd(\mathbf{B}G, \mathbf{B}H):

* * h * g 1 ϕ 1(g 1) ϕ 2(g 1) * * h * g 2 ϕ 1(g 2) ϕ 2(g 2) * * h *. \array{ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast \\ \downarrow^{\mathrlap{g_1}} && && \downarrow^{\mathrlap{\phi_1(g_1)}} && \downarrow^{\mathrlap{\phi_2(g_1)}} \\ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast \\ \downarrow^{\mathrlap{g_2}} && && \downarrow^{\mathrlap{\phi_1(g_2)}} && \downarrow^{\mathrlap{\phi_2(g_2)}} \\ \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast } \,.

The operation of forming π 1\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

π 1:Grpd */Grp. \pi_1 \colon Grpd^{\ast/} \longrightarrow Grp \,.

Restricted to connected groupoids among the pointed groupoids, the functor π 1:Grpd 1 */Grp\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 GG any group then π 1(BG,*)G\pi_1(\mathbf{B}G,\ast) \simeq G.

The more interesting point to notice is that π 1\pi_1 is indeed a fully faithful (2,1)-functor, in that for any (X,x),(Y,y)Grpd 1 */(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1} then the functor

(π 1) X,Y:Grpd */((X,y),(Y,y))Grp(π 1(X,x),π 1(Y,y)) (\pi_1)_{X,Y} \colon Grpd^{\ast/}((X,y),(Y,y)) \longrightarrow Grp(\pi_1(X,x), \pi_1(Y,y))

is an equivalence of hom-groupoids. By prop. 1 it is sufficient to check this for X=BGX = \mathbf{B}G and Y=BHY = \mathbf{B}H with their canonical basepoints, hence to check that for any two groups G,HG,H the functor

(π 1) X,Y:Grpd */((BG,*),(BH,*))Grp(G,H) (\pi_1)_{X,Y} \;\colon\; Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast)) \longrightarrow Grp(G,H)

is an equivalence.

To see this, observe that, by definition of pointed objects via the undercategory under the point, a morphism in Grpd */Grpd^{\ast/} between groupoids of this form B()\mathbf{B}(-) is a diagram in GrpGrp (unpointed) of the form

* h BG Bϕ BH \array{ && \ast \\ & \swarrow &\swArrow_{h}& \searrow \\ \mathbf{B}G && \underset{\mathbf{B}\phi}{\longrightarrow} && \mathbf{B}H }

where the natural isomorphism is equivalently just the choice of an element hHh \in H. Hence these morphisms are pairs (ϕ,h)(\phi,h) of a group homomorphism and an element of the domain.

We claim that the (2,1)-functor π 1\pi_1 takes such (ϕ,h)(\phi,h) to the homomorphism Ad h 1ϕ:GHAd_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H. To see this, consider via remark 1 this functor as forming loops:

π 1(BG,*)={ * * g * BG} gG. \pi_1(\mathbf{B}G,\ast) = \left\{ \array{ && \ast \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{g}} && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}G } \right\}_{g\in G} \,.

This shows that on a morphism as above this acts by forming the pasting

* * g * h BG ϕ BH= * * hϕ(g)h 1 * h BG ϕ BH. \array{ && \ast \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{g}} && \ast \\ & \searrow && \swarrow &\swArrow_{\mathrlap{h}}& \searrow \\ && \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H } \;\;\;\; = \:\;\;\; \array{ && && \ast \\ && & \swarrow && \searrow \\ && \ast && \swArrow_{\mathrlap{h\phi(g)h^{-1}}} && \ast \\ & \swarrow & \swArrow_{\mathrlap{h}} & \searrow && \swarrow \\ \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H } \,.

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 hh are precisely the information that “trivializes” the conjugation action which in prop. 2 prevents the bare set of group homomorphism: a 2-morphism (ϕ 1,h 1)(ϕ 2,h 2)(\phi_1, h_1) \Rightarrow (\phi_2,h_2) in Grp */Grp^{\ast/} is a natural isomorphism of groupoids

BG ϕ 1 BH id h id BG ϕ 2 BH \array{ \mathbf{B}G &\stackrel{\phi_1}{\longrightarrow}& \mathbf{B}H \\ {}^{\mathllap{id}}\downarrow &\Downarrow^{\mathrlap{h}}& \downarrow^{\mathrlap{id}} \\ \mathbf{B}G &\underset{\phi_2}{\longrightarrow}& \mathbf{B}H }

(encoding a conjugation relation ϕ 2=Ad hϕ 1\phi_2 = Ad_{h} \circ \phi_1 as above) such that we have the pasting relation

* h 1 BG ϕ 1 BH id h id BG ϕ 2 BH= * h 2 BG ϕ 2 BH. \array{ && \ast \\ & \swarrow &\swArrow_{h_1}& \searrow \\ \mathbf{B}G && \stackrel{\phi_1}{\longrightarrow} && \mathbf{B}H \\ {}^{\mathllap{id}}\downarrow && \Downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{id}} \\ \mathbf{B}G &&\underset{\phi_2}{\longrightarrow} && \mathbf{B}H } \;\;\;\;\; = \;\;\;\;\; \array{ && \ast \\ & \swarrow &\swArrow_{h_2}& \searrow \\ \mathbf{B}G && \underset{\phi_2}{\longrightarrow} && \mathbf{B}H } \,.

But this says in components that h 2=h 1hh_2 = h_1\cdot h. Hence there is a at most one morphism in Grpd */((BG,*),(BH,*))Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast)) from (ϕ 1,h 1)(\phi_1,h_1) to (ϕ 2,h 2)(\phi_2,h_2): it exists if ϕ 2=Ad hϕ 1\phi_2 = Ad_h \circ \phi_1 and h 2=h 1hh_2 = h_1\cdot h.

But since, by the previous argument, the functor π 1\pi_1 takes (ϕ 1,h 1)(\phi_1,h_1) to Ad h 1 1ϕ 1Ad_{h_1^{-1}} \circ \phi_1, this means that such a morphism exists precisely if both (ϕ 1,h 1)(\phi_1,h_1) and (ϕ 2,h 2)(\phi_2,h_2) are taken to the same group homomorphism by π 1\pi_1

Ad h 2 1ϕ 2=Ad h 1h 1 1ϕ 2=Ad h 1 1ϕ 1. Ad_{h_2^{-1}} \circ \phi_2 = Ad_{h^{-1}\cdot h_1^{-1}}\circ \phi_2 = Ad_{h_1^{-1}} \circ \phi_1 \,.

This establishes that π 1\pi_1 is alspo an equivalence on all hom-groupoids.

This proof also shows that B()\mathbf{B}(-) is in fact the inverse equivalence:


There is an equivalence of (2,1)-categories between pointed connected groupoids and plain groups

GrpBπ 1=Ω *Grpd 1 */ Grp \stackrel{\underoverset{\simeq}{\pi_1 = \Omega_\ast}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}} Grpd^{\ast/}_{\geq 1}

given by forming loop space objects and by forming deloopings.

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

Grp(H)BΩH 1 */ Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}} \mathbf{H}_{\geq 1}^{\ast /}

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 June 10, 2015 14:18:18 by Urs Schreiber (