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

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.

Definition

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

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

Let $S^n \,\coloneqq\, \partial \Delta[n+1]$ (or $S^n \coloneqq Ex^\infty \partial \Delta[n+1]$) 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 $X \in \mathbf{H}$ an object, the power object $X^{S^n} \,\in\, \mathbf{H}$ plays the role of the space of of maps from the $n$-sphere into $X$, as in the definition of simplicial homotopy groups, to which this reduces in the case that $\mathbf{H} =$ ∞Grpd.

Moreover, powering of the canonical morphism $i_n \colon * \to S^n$ induces a morphism

$$X^{i_n} \;\colon\; X^{S^n} \longrightarrow X$$

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

Of objects

Definition

(categorical homotopy groups)

For $n \in \mathbb{N}$ define

$$\pi_n(X) \;\coloneqq\; \tau_{\leq 0} X^{i_n} \;\;\;\in\; \mathbf{H}_{/X}$$

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

Passing to the 0-truncation here amounts to dividing out the homotopies between maps from the $n$-sphere into $X$. The 0-truncated objects in $\mathbf{H}_{/X}$ have the interpretation of sheaves on $X$. 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

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

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

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

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

$$\pi_n(X) \times \pi_n(X) \to \pi_n(X)$$

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

Of morphisms

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

Definition

(homotopy groups of morphisms)

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

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

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

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

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

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

Example

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

Example

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

Properties

In $\infty Grpd$

For the case that $\mathbf{H} =$ ∞Grpd $\simeq$ Top, the $(\infty,1)$-topos theoretic definition of categorical homotopy groups in $\mathbf{H}$ reduces to the ordinary notion of homotopy groups in Top. For $\infty Grpd$ 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 $f : X \to Y$ is equivalent to the following recursive definition

Definition/Proposition

(recursive homotopy groups of morphisms)

For $n \geq 1$ we have

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

This is HTT, remark 6.5.1.3.

This is the generalization of the familiar fact that loop space objects have the same but shifted homotopy groups: In the special case that $X = *$ and $f$ is $f : * \to Y$ we have $X \times_Y X = \Omega_f Y$ and $X \to X \times_Y X$ is just $* \to \Omega_f Y$, so that

$$\pi_n(f) = \pi_n(Y)$$

and

$$\pi_{n-1}(X \to X \times_Y X) \simeq \pi_{n-1} \Omega_f Y \,.$$

Proposition

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

$$\cdots \to f^* \pi_{n+1}(g) \stackrel{\delta_{n+1}}{\to} \pi_n(f) \stackrel{g \circ f}{\to} \to f^* \pi_n(g) \stackrel{\delta_n}{\to} \pi_{n-1}(f) \to \cdots$$

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

This is HTT, remark 6.5.1.5.

Behaviour under geometric morphisms

Proposition

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

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

$$k^* (\pi_n(f)) \simeq \pi_n(k^* f)$$

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

This is HTT, remark 6.5.1.4.

Connected and truncated objects

Let $X \in \mathbf{H}$.

• The object $X$ is $n$-truncated if it is a k-truncated object for some $k \gt n$ and if all its categorical homotopy groups above degree $n$ vanish.

  Every object decomposes as a sequence of $n$-truncated objects: the Postnikov tower in an (∞,1)-category.

• The object $X$ is $n$-connected if the terminal morphism $X \to *$ is an effective epimorphism and if all categorical homotopy groups below degree $n$ are trivial.

• The object $X$ is an Eilenberg-MacLane object of degree $n$ if it is both $n$-connected and $n$-truncated.

Models

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

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

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

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

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

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

• $\bar \pi_n(X) \to \bar \pi_0(X)$ for the sheafification of these presheaves.

Then these $\bar \pi_n(X) \to \bar \pi_0(X)$ are the sheaves of categorical homotopy groups of the object represented by $X$.

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

Related concepts

• long exact sequence of homotopy groups

References

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

• Jacob Lurie, Higher Topos Theory .

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

• J. F. Jardine, Simplicial presheaves (page 4)