OBVIOUS NATURAL MORPHISMS OF SHEAVES ARE UNIQUE

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 · · · f∗g ∗ · · · 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. Commutative diagrams express one of the most typical subtle beauties of mathematics, namely that a single object (in this case an arrow in some category) can be realized by several independent constructions. The more interesting the constructions, the more insight is gained by carefully verifying commutativity, and it is tempting to take the inverse claim to mean that comparable arrows, constructed mundanely, should be expected to be equal and are therefore barely worth proving to be so. As half the purpose of a human-produced publication is to provide insight, there is some validity in thinking that any opportunity to reduce both length and tedium is worth taking, but this is at odd