amazing right adjoint


Differential geometry

differential geometry

synthetic differential geometry






Compact objects



William Lawvere’s definition of an atomic infinitesimal space is as an object Δ\Delta in a topos 𝒯\mathcal{T} such that the inner hom functor () Δ:𝒯𝒯(-)^\Delta : \mathcal{T} \to \mathcal{T} has a right adjoint (is an atomic object).

Notice that by definition of inner hom, () Δ(-)^\Delta 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 𝒯=Sh(C)\mathcal{T} = Sh(C) is a Grothendieck topos, that the Grothendieck topology on the site CC is subcanonical. Let ΔCSh(C)\Delta \in C \hookrightarrow Sh(C) be a representable object.

Then () Δ(-)^\Delta has a right adjoint, hence Δ\Delta is an atomic infinitesimal space, precisely if it preserves colimits.

This is a special case of the general adjoint functor theorem.

For if () Δ(-)^\Delta preserves colimits, its right adjoint is

() Δ:(YSh(C))(USh C(U Δ,Y)). (-)_\Delta : (Y \in Sh(C)) \mapsto (U \mapsto Sh_C(U^\Delta, Y)) \,.

The Y ΔY_\Delta defined this way is indeed a sheaf, due to the assumption that () Δ(-)^\Delta preserves colimits. So this is indeed a right adjoint.

A topos 𝒳\mathcal{X} is a local topos (over Set) if its global section functor Γ=Hom(* 𝒳,)\Gamma = Hom(\ast_{\mathcal{X}}, -) admits a right adjoint. This is hence an “external” version of the amazing right adjoint, exhibiting * 𝒳\ast_{\mathcal{X}} as “atomic”.


Revised on May 28, 2014 14:04:13 by David Roberts (