Tininess and right adjoints to exponentials

TL;DR

This work investigates objects T in cartesian closed categories whose exponential functor (-)^T has a right adjoint (-)_T, a notion tied to infinitesimal or atomic tininess in toposes and SDG contexts. It develops a cohesive theory around atomic objects, including their stability under retracts, a generalized singleton construction in elementary toposes, and interactions with precohesive structures. The results show strong constraints in McLarty toposes (e.g., many atomic objects are connected and contractible) and provide detailed behavior in presheaf and Grothendieck toposes, accompanied by concrete examples and counterexamples. Together, these findings clarify how tininess relates to actual contractibility and connectedness within various topos-theoretic frameworks, informing foundations for Synthetic Differential Geometry and related areas.

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.

Theorems & Definitions (85)

• Lemma 1.1
• proof
• Theorem 1.2
• proof
• Theorem 1.3
• proof
• Proposition 1.4
• proof
• Theorem 1.5
• proof