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

Contents

### Context

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

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

#### Homotopy theory

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

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 $i_n \colo * \to S^n$ 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

The intrinsic $(\infty,1)$-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