totally connected geometric morphism
topos theory Background
Cohomology and homotopy
In higher category theory
Totally connected geometric morphism
geometric morphism is f : E → S f\colon E\to S totally connected if
locally connected, i.e. its inverse image functor has a left adjoint f * f^* which is f ! f_! - S S indexed, and
is f ! f_! left exact, i.e. preserves finite limits.
When thinking of
as a topos over E E via S S , we say that it is a f f totally connected . In particular, when -topos S S and S = Set S=Set is the unique f = ( L Const , Γ ) f = (L Const, \Gamma) global sections geometric morphism, we call a E E totally connected topos. Properties
Of course, any totally connected geometric morphism is
connected, since the terminal object is a particular finite limit. It is also strongly connected, since finite products are also finite limits. Examples
of sheaves on a Sh ( X ) Sh(X) topological space is totally connected iff has a X X dense point (a single point whose closure is all of ). X X
A presheaf topos
is totally connected iff Psh ( C ) Psh(C) is C C cofiltered. Totally connected sites
site is called C C totally connected if
is C C cofiltered, and
covering sieve in is C C connected, when regarded as a subcategory of a slice category.
The second condition implies that all constant presheaves are sheaves, and hence that the left adjoint
of Colim : Psh ( C ) → Set Colim\colon Psh(C) \to Set restricts to Const : Set → Psh ( C ) Const\colon Set\to Psh(C) to give a left adjoint of Sh ( C ) Sh(C) . Cofilteredness of L Const L Const is exactly what is needed for left exactness of C C , essentially by definition. Hence the topos of sheaves on any totally connected site is totally connected. Colim : Psh ( C ) → Set Colim\colon Psh(C) \to Set
Conversely, one can show that any totally connected topos can be (but need not be) presented by some totally connected site.
Chapter C3.6 in
Revised on January 6, 2011 01:10:44