strongly connected topos
Cohomology and homotopy
In higher category theory
A sheaf topos is called strongly connected if it is a locally connected topos
such that the extra left adjoint in addition preserves finite products (the terminal object and binary products).
This means it is in particular also a connected topos.
If preserves even all finite limits then is called a totally connected topos.
If a strongly connected topos is also a local topos, then it is a cohesive topos.
The “strong” in “strongly connected” may be read as referring to being a strong adjunction in that we have a natural isomorphism for the internal homs in the sense that
This follows already for connected and essential if preserves products, because this already implies the equivalent Frobenius reciprocity isomorphism. See here for more.
Revised on December 7, 2011 19:47:10
by Urs Schreiber