amazing right adjoint
objects such that commutes with certain colimits
Bill Lawvere’s definition of an atomic infinitesimal space is as an object in a topos such that the inner hom functor has a right adjoint.
Notice that by definition of inner hom, always has a left adjoint. A right adjoint can only exist for very particular objects. Therefore the term amazing right adjoint
right adjoints to representable exponentials
Assume is a Grothendieck topos, that the Grothendieck topology on the site is subcanonical. Let be a representable object.
Then has a right adjoint, hence is an atomic infinitesimal space, precisely if it preserves colimits.
This is a special case of the general adjoint functor theorem.
For if preserves colimits, its right adjoint is
(-)_\Delta : (Y \in Sh(C)) \mapsto (U \mapsto Sh_C(U^\Delta, Y))
The defined this way is indeed a sheaf, due to the assumption that preserves colimits. So this is indeed a right adjoint.
appendix 4 of
Revised on March 3, 2011 11:27:00
by Urs Schreiber