Obvious natural morphisms of sheaves are unique

Ryan Cohen Reich

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.

Keywords: commutative diagrams, coherence theorem, string diagrams, pullback, pushforward

2010 MSC: Primary 14A15; Secondary 18D30, 18A25

Theory and Applications of Categories, Vol. 29, 2014, No. 4, pp 48-99.

Published 2014-04-15.


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