### Context

#### Differential geometry

differential geometry

synthetic differential geometry

## Applications

#### Compact objects

objects $d\in C$ such that $C\left(d,-\right)$ commutes with certain colimits

# 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 $\left(-{\right)}^{\Delta }:𝒯\to 𝒯$ has a right adjoint.

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

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

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

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

$\left(-{\right)}_{\Delta }:\left(Y\in \mathrm{Sh}\left(C\right)\right)↦\left(U↦{\mathrm{Sh}}_{C}\left({U}^{\Delta },Y\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$(-)_\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 $\left(-{\right)}^{\Delta }$ preserves colimits. So this is indeed a right adjoint.

# References

appendix 4 of

Revised on March 3, 2011 11:27:00 by Urs Schreiber (131.211.232.194)