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arXiv · 1307.4678

Obvious natural morphisms of sheaves are unique

Ryan Cohen Reich
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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.

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Ryan Cohen Reich. 2014-04-06. Obvious natural morphisms of sheaves are unique. https://arxiv.org/abs/1307.4678

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