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amazing right adjoint

Definition

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

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

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

For if $(-{)}^{\Delta }$ preserves colimits, its right adjoint is

$$(-{)}_{\Delta }:(Y\in \mathrm{Sh}(C))\mapsto (U\mapsto {\mathrm{Sh}}_C(U^{\Delta },Y)).$$(-)_\Delta : (Y \in Sh(C)) \mapsto (U \mapsto Sh_C(U^\Delta, Y)) \,.

The $Y_{\Delta }$ defined this way is indeed a sheaf, due to the assumption that $(-{)}^{\Delta }$ preserves colimits. So this is indeed a right adjoint.

References

appendix 4 of

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis