(2,1)-quasitopos?
structures in a cohesive (∞,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.
Recall that since an (∞,1)-topos $\mathbf{H}$ has all limits, it is naturally powered over ∞Grpd:
Let $S^n = \partial \Delta[n+1]$ (or $S^n := Ex^\infty \partial \Delta[n+1]$) 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.
By powering the canonical morphism $i_n : * \to S^n$ induces a morphism
which is restriction to the basepoint. This morphism may be regarded as an object of the over (∞,1)-topos $\mathbf{H}_{/X}$.
(categorical homotopy groups)
For $n \in \mathbb{N}$ define
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 $\mathbf{H}/X$ 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
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
Therefore the cogroup operations $S^n \to S^n \coprod_* S^n$ induce group operations
is the sheaf topos $\tau_{\leq 0} \mathbf{H}_{/X}$. By the usual argument aboiut homotopy groups, these are trivial for $n = 0$ and abelian for $n \geq 2$.
It is frequently useful to speak of homotopy groups of a morphism $f : X \to Y$ in an $(\infty,1)$-topos
(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
in $\mathbf{H}_{/f}$ to
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 $x$.
Examples
If $Y = *$ then there is an essentially unique morphism $f : X \to *$ whose homotopy fiber is $X$ itself. Accordingly $\pi_n(f) \simeq \pi_n(X)$.
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 $\Omega_f Y$.
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.
The definition of the homotopy groups of a morphism $f : X \to Y$ is equivalent to the following recursive definition
(recursive homotopy groups of morphisms)
For $n \geq 1$ we have
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
and
Given a sequence of morphisms $X \stackrel{f}{\to}Y \stackrel{g}{\to} Z$ in $\mathbf{H}$, there is a long exact sequence
in the topos $Disc(\mathbf{H}_{/X})$.
This is HTT, remark 6.5.1.5.
Geometirc 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
in $Disc(\mathbf{H}_{/k^* Y})$.
This is HTT, remark 6.5.1.4.
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.
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
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 thes $\bar \pi_n(X) \to \bar \pi_0(X)$ 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 of
The model in terms of the model structure on simplicial presheaves is duscussed for instance in