(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
An -connected object is an object all whose homotopy groups equal to or below degree are trivial.
More precisely, an object in an ∞-stack (∞,1)-topos is -connected if its categorical homotopy groups equal to or below degree are trivial.
The complementary notion is that of an n-truncated object of an (∞,1)-category.
The Whitehead tower construction produces -connected objects.
This appears as HTT, def. 18.104.22.168, but under the name ”-connective”. Another possible term is ”-simply connected”; see n-connected space for discussion.
One adopts the following convenient terminology.
Every object is -connected.
A -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 -connected morphisms. For more on this see n-connected/n-truncated factorization system.
An object is -connected (for ) precisely if its n-truncation is the terminal object of .
This is HTT, prop. 22.214.171.124.
This is HTT, prop. 126.96.36.199, item 2.
The class of -connected morphisms is stable under pullback and pushout.
If the pullback of a morphism along an effective epimorphism is -connected, then so is the original morphism.
This is HTT, prop. 188.8.131.52, item 6.
A morphism is -connected precisely if it is an effective epimorphism and the diagonal morphism into the (∞,1)-pullback
This appears as HTT, prop. 184.108.40.206.
See also n-connected/n-truncated factorization system.
This appears as a remark in HTT, Example 220.127.116.11. A construction of the factorization in terms of a model category presentation is in (Rezk, prop. 8.5).
The truncated / connected clock
In a hypercomplete (∞,1)-topos the -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 = -connected
effective epimorphism = -connected
0-connected, 1-connected, 2-connected, ;
-connected = equivalence = -truncated
monomorphism = -truncated
0-truncated, 1-truncated, 2-truncated,
-truncated = any morphism
In the the (∞,1)-category Top we have that an object is -connected precisely if it is an n-connected topological space:
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 , being 0-connected is equivalent to being full.
The homotopy pullback is given by the groupoid whose objects are triples and whose morphisms are corresponding tuples of morphisms in making the evident square in commute.
By prop. 3 it is sufficient to check that the diagonal functor is (-1)-connected, hence, as before, essentially surjective, precisely if is full.
First assume that is full. Then for any object, by fullness of there is a morphism in , such that .
Accordingly we have a morphism in
to an object in the diagonal.
Conversely, assume that the diagonal is essentially surjective. Then for every pair of objects such that there is a morphism we are guaranteed morphisms and such that
Therefore is a preimage of under , and hence 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