(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An $n$-connected object is an object all whose homotopy groups equal to or below degree $n$ are trivial.
More precisely, an object in an ∞-stack (∞,1)-topos is $n$-connected if its categorical homotopy groups equal to or below degree $n$ are trivial.
The complementary notion is that of an n-truncated object of an (∞,1)-category.
The Whitehead tower construction produces $n$-connected objects.
An object $X$ in an (∞,1)-topos $\mathbf{H}$ is called $n$-connected for $-1 \leq n \in \mathbb{Z}$ if
the terminal morphism $X \to *$ is an effective epimorphism;
all categorical homotopy groups equal to or below degree $n$ are trivial.
A morphism $f : X \to Y$ in an $(\infty,1)$-topos is called $n$-connected if
regarded as an object in the over-(∞,1)-category $\mathbf{H}_{/Y}$ all categorical homotopy groups equal to or below degree $n$ are trivial.
This appears as HTT, def. 6.5.1.10, but under the name “$(n+1)$-connective”. Another possible term is “$n$-simply connected”; see n-connected space for discussion.
One adopts the following convenient terminology.
Every object is $(-2)$-connected.
A $(-1)$-connected object is also called an inhabited object.
A 0-connected object is simply called a connected object.
Notice that effective epimorphisms are precisely the $(-1)$-connected morphisms. For more on this see n-connected/n-truncated factorization system.
An object $X$ is $n$-connected, def. 1, precisely if its n-truncation $\tau_{\leq n} X$ is the terminal object of $\mathbf{H}$ (hence precisely if it is $\tau_{\leq n}$-comodal).
This is HTT, prop. 6.5.1.12.
Every equivalence is $\infty$-connected.
This is HTT, prop. 6.5.1.16, item 2.
In a general $(\infty,1)$-topos the converse is not true: not every $\infty$-connected morphisms needs to be an equivalence. It is true in a hypercomplete (∞,1)-topos.
The class of $n$-connected morphisms is stable under pullback and pushout.
If the pullback of a morphism along an effective epimorphism is $n$-connected, then so is the original morphism.
This is HTT, prop. 6.5.1.16, item 6.
A morphism $f : X \to Y$ is $n$-connected precisely if it is an effective epimorphism and the diagonal morphism into the (∞,1)-pullback
is $(n-1)$-connected.
This appears as HTT, prop. 6.5.1.18.
Let $\mathbf{H}$ be an (∞,1)-topos. For all $(-2) \leq n \leq \infty$ the class of $n$-connected morphisms in $\mathbf{H}$ forms the left class in a orthogonal factorization system in an (∞,1)-category. The right class is that of n-truncated morphisms in $\mathbf{H}$.
See also n-connected/n-truncated factorization system.
This appears as a remark in HTT, Example 5.2.8.16. A construction of the factorization in terms of a model category presentation is in (Rezk, prop. 8.5).
In a hypercomplete (∞,1)-topos the $\infty$-connected morphisms are precisely the equivalences.
Therefore in such a context we have the following “clock” of notions of truncated object in an (infinity,1)-category / connected :
any morphism = $(-2)$-connected
effective epimorphism = $(-1)$-connected
0-connected, 1-connected, 2-connected, $\cdots$;
$\infty$-connected = equivalence = $(-2)$-truncated
monomorphism = $(-1)$-truncated
0-truncated, 1-truncated, 2-truncated, $\cdots$
$\infty$-truncated = any morphism
In the the (∞,1)-category Top we have that an object is $n$-connected precisely if it is an n-connected topological space:
a $(-1)$-connected object is an inhabited space.
a $0$-connected object is a path-connected space.
a $1$-connected object is a simply connected space.
a $\infty$-connected object is a contractible space.
Let $f : X \to Y$ be a functor between groupoids. Regarded as a morphism in ∞Grpd $f$ is 0-connected precisely if it is an essentially surjective and full functor.
As discussed there, an effective epimorphism in ∞Grpd between 1-groupoids is precisely an essentially surjective functor.
So it remains to check that for an essentially surjective $f$, being 0-connected is equivalent to being full.
The homotopy pullback $X \times_Y X$ is given by the groupoid whose objects are triples $(x_1, x_2 \in X, \alpha : f(x_1) \to f(x_2))$ and whose morphisms are corresponding tuples of morphisms in $X$ making the evident square in $Y$ commute.
By prop. 3 it is sufficient to check that the diagonal functor $X \to X \times_Y X$ is (-1)-connected, hence, as before, essentially surjective, precisely if $f$ is full.
First assume that $f$ is full. Then for $(x_1,x_2, \alpha) \in X \times_Y X$ any object, by fullness of $f$ there is a morphism $\hat \alpha : x_1 \to x_2$ in $X$, such that $f(\hat \alpha) = \alpha$.
Accordingly we have a morphism $(\hat \alpha,id) : (x_1, x_2) \to (x_2, x_2)$ in $X \times_Y X$
to an object in the diagonal.
Conversely, assume that the diagonal is essentially surjective. Then for every pair of objects $x_1, x_2 \in X$ such that there is a morphism $\alpha : f(x_1) \to f(x_2)$ we are guaranteed morphisms $h_1 : x_1 \to x_2$ and $h_2 : x_2 \to x_2$ such that
Therefore $h_2^{-1}\circ h_1$ is a preimage of $\alpha$ under $f$, and hence $f$ is full.
See also (eso+full, faithful) factorization system.
Section 6.5.1 of
A discussion in terms of model category presentations is in section 8 of