Tininess and right adjoints to exponentials

TL;DR

This paper investigates the 'tininess' of objects T in toposes for which the exponential functor (-)^T has a right adjoint, analyzing their categorical and topological properties to understand their foundational significance.

Contribution

It explores the categorical and topological properties of objects T with right adjoints to exponentials, providing insights into their 'tininess' and relevance in synthetic differential geometry.

Findings

Objects T are 'amazingly tiny' in certain toposes.

Categorical behavior of T aligns with Lawvere's 'A.T.O.M.' concept.

Topological properties of T support their foundational role.

Abstract

Objects $T$ whose exponential functor $(-)^T$ admits a right adjoint $(-)_T$ are known under different names. The fact that they exist, yet that the only set that satisfies this in the category of sets is the singleton made Lawvere suggest they ought to be ``amazingly tiny'' -- hence Lawvere's acronym ``A.T.O.M.'' This report explores how intuitively tiny any such object is. Evidences both in favor and to the contrary are produced by looking at their categorical behavior (subobjects, quotients, retracts, etc) when the ambient category is a topos. The topological behavior (connectedness, contractibility, connected components, etc) of both $T$ and $(-)_T$ is further analyzed in toposes that satisfy certain precohesive conditions over their decidable objects, where this tininess is tested against parts of Lawvere's foundational proposal for Synthetic Differential Geometry.