Obvious natural morphisms of sheaves are unique

TL;DR

The paper proves the uniqueness of certain natural transformations between sheaf functors in algebraic geometry, establishing a coherence theorem under specific axioms for geofibered categories.

Contribution

It introduces axioms for geofibered categories that guarantee the uniqueness of natural transformations involving sheaf functors, ensuring diagram commutativity.

Findings

Uniqueness of natural transformations in sheaf-theoretic contexts

Identification of axioms for coherence in geofibered categories

Applicability to standard algebraic geometry sheaf theories

Abstract

We prove that a large class of natural transformations (consisting roughly of those constructed via composition from the "functorial" or "base change" transformations) between two functors of the form $\cdots f^* g_* \cdots$ actually has only one element, and thus that any diagram of such maps necessarily commutes. We identify the precise axioms defining what we call a "geofibered category" that ensure that such a coherence theorem exists. Our results apply to all the usual sheaf-theoretic contexts of algebraic geometry. The analogous result that would include any other of the six functors remains unknown.