(∞,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
Paths and cylinders
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 has all limits, it is naturally powered over ∞Grpd:
Let (or ) be the (Kan fibrant replacement) of the boundary of the (n+1)-simplex, i.e. the model in ∞Grpd of the pointed -sphere.
Then for an object, the power object plays the role of the space of of maps from the -sphere into , as in the definition of simplicial homotopy groups, to which this reduces in the case that ∞Grpd.
By powering the canonical morphism induces a morphism
which is restriction to the basepoint. This morphism may be regarded as an object of the over (∞,1)-topos .
(categorical homotopy groups)
to be the 0-truncation of the object .
Passing to the 0-truncation here amounts to dividing out the homotopies between maps from the -sphere into . The 0-truncated objects in have the interpretation of sheaves on . 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 preserves such finite products so that also
Therefore the cogroup operations induce group operations
is the sheaf topos . By the usual argument aboiut homotopy groups, these are trivial for and abelian for .
It is frequently useful to speak of homotopy groups of a morphism in an -topos
(homotopy groups of morphisms)
For a morphism in an (∞,1)-topos , its homotopy groups are the homotopy groups in the above sense of regarded as an object of the over (∞,1)-category .
So the homotopy sheaf of a morphism is an object of the over (∞,1)-category . This in turn is equivalent to by the map that sends an object
The intuition is that the homotopy sheaf over a basepoint is the homotopy group of the homotopy fiber of containing at .
If then there is an essentially unique morphism whose homotopy fiber is itself. Accordingly .
If then the morphism is a point in and the single homotopy fiber of is the loop space object .
For the case that ∞Grpd Top, the -topos theoretic definition of categorical homotopy groups in reduces to the ordinary notion of homotopy groups in Top. For 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 is equivalent to the following recursive definition
(recursive homotopy groups of morphisms)
For we have
This is HTT, remark 188.8.131.52.
This is the generalization of the familiar fact that loop space objects have the same but shifted homotopy groups: In the special case that and is we have and is just , so that
Given a sequence of morphisms in , there is a long exact sequence
in the topos .
This is HTT, remark 184.108.40.206.
Behaviour under geometric morphisms
Geometirc morphisms of -topos preserve homotopy groups.
If is a geometric morphism of -toposes then for any morphism in there is a canonical isomorphism
This is HTT, remark 220.127.116.11.
Connected and truncated objects
When the (∞,1)-topos is presented by a model structure on simplicial presheaves , then since this is an sSet-enriched model category structure the powering by is modeled, as described at, -limit – Tensoring – Models by the ordinary powering
which is just objectwise the internal hom in sSet. Therefore the -topos theoretical homotopy sheaves of an object in are given by the following construction:
For a presheaf, write
for the presheaf of connected components;
for the presheaf of simplicial homotopy groups with ;
for the sheafification of these presheaves.
Then thes are the sheaves of categorical homotopy groups of the object represented by .
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 -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)