amazing right adjoint
objects such that commutes with certain colimits
William 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 (is an atomic object).
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
The defined this way is indeed a sheaf, due to the assumption that preserves colimits. So this is indeed a right adjoint.
A topos is a local topos (over Set) if its global section functor admits a right adjoint. This is hence an “external” version of the amazing right adjoint, exhibiting as “atomic”.
Revised on May 28, 2014 14:04:13
by David Roberts