(∞,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
Every (∞,1)-topos has a shape . When is locally ∞-connected then this is a genuine ∞-groupoid ∞Grpd. We may think of this as the fundamental ∞-groupoid of the -topos regarded as a generalized space.
But also every locally ∞-connected (∞,1)-topos has an internal notion of fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos for objects of , denoted . (If in addition is fully faithful then we call the shape modality.) Applied to its terminal object this does agree with the fundamental ∞-groupoid of the topos:
Conversely, for an object , the fundamental ∞-groupoid internal to can be identified with the fundamental ∞-groupoid of the locally ∞-connected (∞,1)-topos .
Let be a locally -connected -topos and an object. Then also the over-(∞,1)-topos is locally -connected (as discussed there).
We have then two different definitions of the fundamental -groupoid of : once as – the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos – and once as .
Since is the terminal object in we have by definition
Now observe that since the terminal global section geometric morphism of the over-topos is
and that in the etale geometric morphism is the projection map that sends to .
Let denote the full sub-(∞,1)-category of (∞,1)Topos determined by the locally ∞-connected objects.
The -category (as the category of local homeomorphisms over ) is reflective in ,
with the reflector given by forming the fundamental -groupoid.
Any ∞-groupoid gives rise to an (∞,1)-presheaf (∞,1)-topos , which by the (∞,1)-Grothendieck construction is equivalent to the over-(∞,1)-topos . The -toposes of this form are, by definition, those for which the unique (∞,1)-geometric morphism to is a local homeomorphism of toposes. This construction embeds as a full sub-(∞,1)-category of (∞,1)Topos:
since in particular the -toposes are locally ∞-connected.
To show that is a left adjoint (∞,1)-functor to we demonstrate a natural hom-equivalence
for and .
At shape of an (∞,1)-topos it is shown that we have a natural equivalence
Now observe that furthermore we have a sequence of natural equivalences
More generally the shape of an (∞,1)-topos of reproduces the shape theory of .