Context

Differential geometry

differential geometry

synthetic differential geometry

Applications

Compact objects

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

Contents

Definition

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

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

Assume $\mathcal{T} = Sh(C)$ is a Grothendieck topos, that the Grothendieck topology on the site $C$ is subcanonical. Let $\Delta \in C \hookrightarrow Sh(C)$ be a representable object. Then, $(-)^\Delta: Sh(C)\to Sh(C)$ has a right adjoint, hence $\Delta$ is an atomic infinitesimal space, precisely if $(-)^\Delta: C\to C$ preserves colimits. This is a special case of the general adjoint functor theorem. For if $(-)^\Delta$ preserves colimits, its right adjoint is

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

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.

A topos $\mathcal{X}$ is a local topos (over Set) if its global section functor $\Gamma = Hom(1_{\mathcal{X}}, -)$ admits a right adjoint. This is hence an “external” version of the amazing right adjoint, exhibiting $1_{\mathcal{X}}$ as “atomic”.

References

The ubiquity of right adjoints to exponential functors in the context of synthetic differential geometry was first pointed out in Lawvere (1980). Lawvere (2004) suggests to augment lambda calculus with such fractional operators. Thorough discussion of the concept is in Yetter (1987) and Kock&Reyes (1999). Moerdijk&Reyes (1991) have a succinct overview in the context of SDG as does Lawvere (1997).

Revised on July 5, 2016 10:55:39 by Daniel Luckhardt (134.76.82.216)