A topos may be thought of as a generalized topological space. Accordingly, the notions of
have analogs for toposes, (n,1)-toposes and (∞,1)-toposes
locally connected topos
locally simply connected (2,1)-topos?
locally -connected -topos
locally -connected -topos
The numbering mismatch is traditional from topology; see n-connected space. It reads a bit better if we say locally -simply connected for locally -connected, since locally -(simply) connected is locally simply connected, but being locally -simply connected is still a property of an -topos.
A (∞,1)-sheaf (∞,1)-topos is called locally -connected if the (essentially unique) global section (∞,1)-geometric morphism
extends to an essential geometric morphism -geometric morphism, i.e. there is a further left adjoint
If in addition preserves the terminal object we say that is an ∞-connected (∞,1)-topos.
If preserves even all finite (∞,1)-products we say that is a strongly ∞-connected (∞,1)-topos.
If preserves even all finite (∞,1)-limits we say that is a totally ∞-connected (∞,1)-topos.
Similarly we have:
For an -topos is called locally -connected if the (essentially unique) global section geometric morphism is has an extra left adjoint.
For this reproduces the case of a locally connected topos.
Over locally -connected sites
The follow proposition gives a large supply of examples.
See locally ∞-connected (∞,1)-site/∞-connected (∞,1)-site for the proof.
This includes the following examples.
Over locally -connected topological spaces
By the same kind of argument:
For and for a locally -connected topological space, is a locally -connected -topos.
Locally -connected over--toposes
For a locally -connected -topos, also all its objects are locally -connected, in that their petit over-(∞,1)-toposes are locally -connected.
The two notions of fundamental -groupoids of induced this way do agree, in that there is a natural equivalence
By the general facts recalled at etale geometric morphism we have a composite essential geometric morphism
and is given by sending to .
By the discussion there we need to check that preserves the terminal object:
Relation to slicing
Let be an -topos and a collection of objects such that
Then also itself is locally -connected.
This appears as (Lurie, corollary A.1.7).
Relation to locally connected toposes
For a locally -connected -topos, its underlying (1,1)-topos is a locally connected topos. Moreover, if is strongly connected (the extra left adjoint preserves finite products), then so is .
The global sections geometric morphism is given by homming out of the terminal object and hence preserves 0-truncated objects by definition. Also, by the -adjunction so does : for every and every we have
Therefore by essential uniqueness of adjoints the restriction has a left adjoint given by
Finally, by the discussion here, preserves finite limits. Hence does so if does.
The fact that the terminal geometric morphism is essential gives rise to various induced structures of interest. For instance it induces a notion of
For a more exhaustive list of extra structures see cohesive (∞,1)-topos.
Some discussion of the homotopy category of locally -connected -toposes is around proposition 2.18 of
Under the term locally constant shape the notion appears in section A.1 of
For related references see