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

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

## Theorems

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.

# Contents

## Definition

Recall that since an (∞,1)-topos $H$ has all limits, it is naturally powered over ∞Grpd:

$\left(-{\right)}^{\left(-\right)}:\infty {\mathrm{Grpd}}^{\mathrm{op}}×H\to H\phantom{\rule{thinmathspace}{0ex}}.$(-)^{(-)} : \infty Grpd^{op} \times \mathbf{H} \to \mathbf{H} \,.

Let ${S}^{n}=\partial \Delta \left[1\right]$ (or ${S}^{n}:={\mathrm{Ex}}^{\infty }\partial \Delta \left[n+1\right]$) 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 $X\in H$ an object, the power object ${X}^{{S}^{n}}\in H$ 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 $H=$ ∞Grpd.

By powering the canonical morphism ${i}_{n}:*\to {S}^{n}$ induces a morphism

${X}^{{i}_{n}}:{X}^{{S}^{n}}\to X$X^{i_n} : X^{S^n} \to X

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

### Of objects

###### Definition

(categorical homotopy groups)

For $n\in ℕ$ define

${\pi }_{n}\left(X\right):={\tau }_{\le 0}{X}^{{i}_{n}}\in {H}_{/X}$\pi_n(X) := \tau_{\leq 0} X^{i_n} \in \mathbf{H}_{/X}

to be the 0-truncation of the 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 $H/X$ 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

${X}^{{S}^{k}\coprod _{*}{S}^{l}}\simeq {X}^{{S}_{k}}{×}_{X}{X}^{{S}_{l}}\phantom{\rule{thinmathspace}{0ex}}.$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 ${\tau }_{\le n}:H\to H$ preserves such finite products so that also

${\tau }_{\le 0}{X}^{*\to {S}^{k}\coprod _{*}{S}^{l}}\simeq \left({\tau }_{\le 0}{X}^{*\to {S}^{k}}\right)×\left({\tau }_{\le 0}\right){X}^{*\to {S}^{k}}\phantom{\rule{thinmathspace}{0ex}}.$\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

${\pi }_{n}\left(X\right)×{\pi }_{n}\left(X\right)\to {\pi }_{n}\left(X\right)$\pi_n(X) \times \pi_n(X) \to \pi_n(X)

is the sheaf topos ${\tau }_{\le 0}{H}_{/X}$. By the usual argument aboiut homotopy groups, these are trivial for $n=0$ and abelian for $n\ge 2$.

### Of morphisms

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

###### Definition

(homotopy groups of morphisms)

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

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

$\begin{array}{ccc}& & Q\\ & ↙& & ↘\\ X& & \stackrel{f}{\to }& & Y\end{array}$\array{ && Q \\ & \swarrow && \searrow \\ X &&\stackrel{f}{\to}&& Y }

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

$\begin{array}{ccc}& & Q\\ & ↙\\ X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && Q \\ & \swarrow \\ X } \,.

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

Examples

If $Y=*$ then there is an essentially unique morphism $f:X\to *$ whose homotopy fiber is $X$ itself. Accordingly ${\pi }_{n}\left(f\right)\simeq {\pi }_{n}\left(X\right)$.

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 $\infty \mathrm{Grpd}$

For the case that $H=$ ∞Grpd $\simeq$ Top, the $\left(\infty ,1\right)$-topos theoretic definition of categorical homotopy groups in $H$ reduces to the ordinary notion of homotopy groups in Top. For $\infty \mathrm{Grpd}$ 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 $n\ge 1$ we have

${\pi }_{n}\left(f\right)\simeq {\pi }_{n-1}\left(X\to X{×}_{Y}X\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in \mathrm{Disc}\left({H}_{/X}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $X{×}_{Y}X={\Omega }_{f}Y$ and $X\to X{×}_{Y}X$ is just $*\to {\Omega }_{f}Y$, so that

${\pi }_{n}\left(f\right)={\pi }_{n}\left(Y\right)$\pi_n(f) = \pi_n(Y)

and

${\pi }_{n-1}\left(X\to X{×}_{Y}X\right)\simeq {\pi }_{n-1}{\Omega }_{f}Y\phantom{\rule{thinmathspace}{0ex}}.$\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 $H$, there is a long exact sequence

$\cdots \to {f}^{*}{\pi }_{n+1}\left(g\right)\stackrel{{\delta }_{n+1}}{\to }{\pi }_{n}\left(f\right)\stackrel{g\circ f}{\to }\to {f}^{*}{\pi }_{n}\left(g\right)\stackrel{{\delta }_{n}}{\to }{\pi }_{n-1}\left(f\right)\to \cdots$\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 $\mathrm{Disc}\left({H}_{/X}\right)$.

This is HTT, remark 6.5.1.5.

### Behaviour under geometric morphisms

###### Proposition

Geometirc morphisms of $\left(\infty ,1\right)$-topos preserve homotopy groups.

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

${k}^{*}\left({\pi }_{n}\left(f\right)\right)\simeq {\pi }_{n}\left({k}^{*}f\right)$k^* (\pi_n(f)) \simeq \pi_n(k^* f)

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

This is HTT, remark 6.5.1.4.

### Connected and truncated objects

Let $X\in H$.

• The object $X$ is $n$-truncated if it is a k-truncated object for some $k>n$ 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 $H$ is presented by a model structure on simplicial presheaves $\left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{loc}}$, then since this is an sSet-enriched model category structure the powering by $\infty \mathrm{Grpd}$ is modeled, as described at, $(\infty,1)$-limit -- Tensoring -- Models by the ordinary powering

${\mathrm{sSet}}^{\mathrm{op}}×\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\to \left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\phantom{\rule{thinmathspace}{0ex}},$sSet^{op} \times [C^{op}, sSet] \to [C^{op}, sSet] \,,

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

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

• ${\pi }_{0}\left(X\right)\in \left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ for the presheaf of connected components;

• ${\pi }_{n}\left(X\right)={\coprod }_{\left[x\right]\in {\pi }_{0}\left(X\right)}{\pi }_{n}\left(X,x\right)$ for the presheaf of simplicial homotopy groups with $n\ge 1$;

• ${\overline{\pi }}_{n}\left(X\right)\to {\overline{\pi }}_{0}\left(X\right)$ for the sheafification of these presheaves.

Then thes ${\overline{\pi }}_{n}\left(X\right)\to {\overline{\pi }}_{0}\left(X\right)$ 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

## References

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

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

• Jardine, Simplicial presheaves (page 4)

Revised on September 7, 2011 15:12:16 by Urs Schreiber (82.113.99.42)