nLab categorical homotopy groups in an (infinity,1)-topos



(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos

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


This is a sub-entry of homotopy groups in an (∞,1)-topos.

For the other notion of homotopy groups see geometric homotopy groups in an (∞,1)-topos.



Since an (∞,1)-topos H\mathbf{H} has all ( , 1 ) (\infty,1) -limits, it is powered over ∞Grpd (see at Powering of \infty-toposes over \infty-groupoids):

() ():Grpd op×HH. (-)^{(-)} \;\colon\; \infty Grpd^{op} \times \mathbf{H} \longrightarrow \mathbf{H} \,.

Let S nΔ[n+1]S^n \,\coloneqq\, \partial \Delta[n+1] (or S nEx Δ[n+1]S^n \coloneqq Ex^\infty \partial \Delta[n+1]) be the (Kan fibrant replacement) of the boundary of the (n+1)-simplex, i.e. the model in ∞Grpd of the pointed n-sphere.

Then for XHX \in \mathbf{H} an object, the power object X S nHX^{S^n} \,\in\, \mathbf{H} plays the role of the space of of maps from the nn-sphere into XX, as in the definition of simplicial homotopy groups, to which this reduces in the case that H=\mathbf{H} = ∞Grpd.

Moreover, powering of the canonical morphism i ncolo*S n i_n \colo * \to S^n induces a morphism

X i n:X S nX X^{i_n} \;\colon\; X^{S^n} \longrightarrow X

which is restriction to the basepoint. This morphism may be regarded as an object of the slice (∞,1)-topos H /X\mathbf{H}_{/X}.

Of objects


(categorical homotopy groups)

For nn \in \mathbb{N} define

π n(X)τ 0X i nH /X \pi_n(X) \;\coloneqq\; \tau_{\leq 0} X^{i_n} \;\;\;\in\; \mathbf{H}_{/X}

to be the 0-truncation of the power object X i nX^{i_n}.

Passing to the 0-truncation here amounts to dividing out the homotopies between maps from the nn-sphere into XX. The 0-truncated objects in H /X\mathbf{H}_{/X} have the interpretation of sheaves on XX. So in the world of ∞-stacks a homotopy group object is a sheaf of groups.

To see that there is indeed a group structure on these homotopy sheaves as usual, notice from the general properties of powering we have that

X S k *S lX S k× XX S l. X^{S^k \coprod_* S^l} \simeq X^{S_k} \times_X X^{S_l} \,.

From the discussion of properties of truncation we have that τ n:HH\tau_{\leq n} : \mathbf{H} \to \mathbf{H} preserves such finite products so that also

τ 0X *S k *S l(τ 0X *S k)×(τ 0)X *S k. \tau_{\leq 0} X^{* \to S^k \coprod_* S^l} \simeq (\tau_{\leq 0} X^{* \to S^k} ) \times (\tau_{\leq 0}) X^{* \to S^k} \,.

Therefore the cogroup operations S nS n *S nS^n \to S^n \coprod_* S^n induce group operations

π n(X)×π n(X)π n(X) \pi_n(X) \times \pi_n(X) \to \pi_n(X)

in the sheaf topos τ 0H /X\tau_{\leq 0} \mathbf{H}_{/X}. By the usual argument about homotopy groups, these are trivial for n=0n = 0 and abelian for n2n \geq 2.

Of morphisms

It is frequently useful to speak of homotopy groups of a morphism f:XYf : X \to Y in an (,1)(\infty,1)-topos


(homotopy groups of morphisms)

For f:XYf : X \to Y a morphism in an (∞,1)-topos H\mathbf{H}, its homotopy groups are the homotopy groups in the above sense of ff regarded as an object of the over (∞,1)-category H /Y\mathbf{H}_{/Y}.

So the homotopy sheaf π n(f)\pi_n(f) of a morphism ff is an object of the over (∞,1)-category Disc((H /Y) /f)Disc(H /f)Disc((\mathbf{H}_{/Y})_{/f}) \simeq Disc(\mathbf{H}_{/f}). This in turn is equivalent to H /X\cdots \simeq \mathbf{H}_{/X} by the map that sends an object

Q X f Y \array{ && Q \\ & \swarrow && \searrow \\ X &&\stackrel{f}{\to}&& Y }

in H /f\mathbf{H}_{/f} to

Q X. \array{ && Q \\ & \swarrow \\ X } \,.

The intuition is that the homotopy sheaf π n(f)Disc(H /X)\pi_n(f) \in Disc(\mathbf{H}_{/X}) over a basepoint x:*Xx : * \in X is the homotopy group of the homotopy fiber of ff containing xx at xx.


If Y=*Y = * then there is an essentially unique morphism f:X*f : X \to * whose homotopy fiber is XX itself. Accordingly π n(f)π n(X)\pi_n(f) \simeq \pi_n(X).


If X=*X = * then the morphism f:*Yf : * \to Y is a point in YY and the single homotopy fiber of ff is the loop space object Ω fY\Omega_f Y.


In Grpd\infty Grpd

For the case that H=\mathbf{H} = ∞Grpd \simeq Top, the (,1)(\infty,1)-topos theoretic definition of categorical homotopy groups in H\mathbf{H} reduces to the ordinary notion of homotopy groups in Top. For Grpd\infty Grpd modeled by Kan complexes or the standard model structure on simplicial sets, it reduces to the ordinary definition of simplicial homotopy groups.

Of homotopy groups of morphisms

The definition of the homotopy groups of a morphism f:XYf : X \to Y is equivalent to the following recursive definition


(recursive homotopy groups of morphisms)

For n1n \geq 1 we have

π n(f)π n1(XX× YX)Disc(H /X). \pi_n(f) \simeq \pi_{n-1}(X \to X \times_Y X) \;\;\; \in Disc(\mathbf{H}_{/X}) \,.

This is HTT, remark

This is the generalization of the familiar fact that loop space objects have the same but shifted homotopy groups: In the special case that X=*X = * and ff is f:*Yf : * \to Y we have X× YX=Ω fYX \times_Y X = \Omega_f Y and XX× YXX \to X \times_Y X is just *Ω fY* \to \Omega_f Y, so that

π n(f)=π n(Y) \pi_n(f) = \pi_n(Y)


π n1(XX× YX)π n1Ω fY. \pi_{n-1}(X \to X \times_Y X) \simeq \pi_{n-1} \Omega_f Y \,.

Given a sequence of morphisms XfYgZX \stackrel{f}{\to}Y \stackrel{g}{\to} Z in H\mathbf{H}, there is a long exact sequence

f *π n+1(g)δ n+1π n(f)gff *π n(g)δ nπ n1(f) \cdots \to f^* \pi_{n+1}(g) \stackrel{\delta_{n+1}}{\to} \pi_n(f) \stackrel{g \circ f}{\to} \to f^* \pi_n(g) \stackrel{\delta_n}{\to} \pi_{n-1}(f) \to \cdots

in the topos Disc(H /X)Disc(\mathbf{H}_{/X}).

This is HTT, remark

Behaviour under geometric morphisms


Geometric morphisms of (,1)(\infty,1)-topos preserve homotopy groups.

If k:HKk : \mathbf{H} \to \mathbf{K} is a geometric morphism of (,1)(\infty,1)-toposes then for f:XYf : X \to Y any morphism in H\mathbf{H} there is a canonical isomorphism

k *(π n(f))π n(k *f) k^* (\pi_n(f)) \simeq \pi_n(k^* f)

in Disc(H /k *Y)Disc(\mathbf{H}_{/k^* Y}).

This is HTT, remark

Connected and truncated objects

Let XHX \in \mathbf{H}.


When the (∞,1)-topos H\mathbf{H} is presented by a model structure on simplicial presheaves [C op,sSet] loc[C^{op}, sSet]_{loc}, then since this is an sSet-enriched model category structure the powering by Grpd\infty Grpd is modeled, as described at, ( , 1 ) (\infty,1) -limit – Tensoring – Models by the ordinary powering

sSet op×[C op,sSet][C op,sSet], sSet^{op} \times [C^{op}, sSet] \to [C^{op}, sSet] \,,

which is just objectwise the internal hom in sSet. Therefore the (,1)(\infty,1)-topos theoretical homotopy sheaves of an object in ([C op,sSet] loc) ([C^{op}, sSet]_{loc})^\circ are given by the following construction:

For X[C op,sSet]X \in [C^{op}, sSet] a presheaf, write

  • π 0(X)[C op,Set]\pi_0(X) \in [C^{op},Set] for the presheaf of connected components;

  • π n(X)= [x]π 0(X)π n(X,x)\pi_n(X) = \coprod_{[x] \in \pi_0(X)} \pi_n(X,x) for the presheaf of simplicial homotopy groups with n1n \geq 1;

  • π¯ n(X)π¯ 0(X)\bar \pi_n(X) \to \bar \pi_0(X) for the sheafification of these presheaves.

Then these π¯ n(X)π¯ 0(X)\bar \pi_n(X) \to \bar \pi_0(X) are the sheaves of categorical homotopy groups of the object represented by XX.

This definition of homotopy sheaves of simplicial presheaves is familiar from the Joyal-Jardine local model structure on simplicial presheaves. See for instance page 4 of Jard07.

this needs more discussion


The intrinsic (,1)(\infty,1)-theoretic description is the topic of section 6.5.1 in:

The model in terms of the model structure on simplicial presheaves is duscussed for instance in

Last revised on October 18, 2021 at 10:36:39. See the history of this page for a list of all contributions to it.