(∞,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
If an (∞,1)-topos is that of (∞,1)-sheaves on (the site of open subsets of) a paracompact topological space – – then its shape is the strong shape of in the sense of shape theory: a pro-object in the category of CW-complexes.
It turns out that may be extracted in a canonical fashion from just the (∞,1)-topos , and in a way that makes sense for any (∞,1)-topos. This then gives a definition of shape of general -toposes.
The composite (∞,1)-functor
is the shape functor . Its value
on an -topos is the shape of .
(∞,1)Topos is the (∞,1)-category of (∞,1)-toposes;
∞Grpd is the -category of ∞-groupoids;
is the (∞,1)-Yoneda embedding;
is the (∞,1)-category of (∞,1)-functors;
is the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those which are left exact functors (preserve finite (∞,1)-limits) and also accessible.
is the functor that produces the (∞,1)-category of (∞,1)-presheaves on (equivalently on the equivalent opposite ∞-groupoid );
is the (∞,1)-category of pro-objects in .
That this does indeed land in accessible left exact functors is shown below.
Notice that for every (∞,1)-topos there is a unique geometric morphism
where ∞Grpd is the -topos of ∞-groupoids, is the global sections (∞,1)-functor and is the constant ∞-stack functor.
The shape of is the composite functor
regarded as an object
For ∞Grpd we have by the (∞,1)-Grothendieck construction-theorem and using that up to equivalence every morphism of -groupoids is a Cartesian fibration (see there) that
is the over-(∞,1)-category. Moreover, by the theorem about limits in ∞Grpd we have that the terminal geometric morphism is the canonical projection . This means that it is an etale geometric morphism. So for any geometric morphism we have a system of adjoint (∞,1)-functors
whose composite is the global section geometric morphism as indicated, because that is terminal.
Notice that in there is a canonical morphism
The image of this under is (using that this preserves the terminal object) a morphism
Conversely, given a morphism of the form in we obtain the base change geometric morphism
One checks that these constructions establish an equivalence
Using this, we see that
Shape of a locally -connected topos
Suppose that is locally ∞-connected, meaning that has a left adjoint which constructs the homotopy ∞-groupoids of objects of . Then is represented by , for we have the following sequence of natural equivalences of ∞-groupoids:
Thus, if we regard as “the fundamental ∞-groupoid of ” — which is reasonable since when consists of sheaves on a locally contractible topological space , is equivalent to the usual fundamental ∞-groupoid of — then we can regard the shape of an -topos as a generalized version of the “homotopy -groupoid” which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of “pro--groupoids.”
It follows also that is not only locally ∞-connected but also ∞-connected, then it has the shape of a point.
Shape of a topological space
For a discussion of how the -topos theoretic shape of relates to the ordinary shape-theoretic strong shape of the topological space see shape theory.
Shape of an essential retract
The following is trivial to observe, but may be useful to note.
Let be an essential geometric morphism of -toposes that exhibits as an essential retract of in that
Then the shape of is equivalent to that of .
Since is the terminal object in the category of Grothendieck -toposes and geometric morphisms, we have
over has the shape of the point.
The definition of shape of -toposes as is due to
This and the relation to shape theory, more precisely the strong shape, of topological spaces is further discussed in section 7.1.6 of