nLab amazing right adjoint

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.

Right adjoints to representable exponentials

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#in_toposes. 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.

Related concepts

• 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”.

• The topic of amazing right adjoints appears to have been studied mostly for toposes, but a related definition has been given for any category (with products): if $O$ is an exponentiable object of a category $\mathcal{C}$, then $O$ being a tiny object in the sense of Definition 0.1 means that the endofunctor $(-)^O$ has a right adjoint. (This adjoint is sometimes denoted $(-)^{1/O}$, c.f. Lawvere, p.269, and called ‘fractional exponential’)

In this situation, one then has, symbolically

$$(-)\times O \quad \dashv\quad (-)^O \quad \dashv\quad (-)^{1/O}$$

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).

• William Lawvere, Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body, Cah.Top.Géom.Diff.Cat. 21 no.4 (1980) pp.377-392. (pdf)

• William Lawvere, Toposes of laws of motion, 1997.

• William Lawvere, Left and right adjoint operations on spaces and data types , Theor. Comp. Sci. 316 (2004) pp.105-111.

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis , Springer Heidelberg 1991. (appendix 4)

• Anders Kock, Gonzalo E. Reyes, Aspects of Fractional Exponent Functors , TAC 5 (1999) pp.251-265. (pdf)

• David Yetter, On Right Adjoints of Exponential Functors , JPAA 45 (1987) pp.287-304. (Corrections in JPAA 58 (1989) pp.103-105)

• William Lawvere, Categorical algebra for continuum micro physics, Journal of Pure and Applied Algebra 175 (2002) 267–287