Geodesic
Obvious natural morphisms of sheaves are unique
Theory and applications of categories, Tome 29 (2014), pp. 48-99.

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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.
Publié le :
Classification : Primary 14A15, Secondary 18D30, 18A25
Keywords: commutative diagrams, coherence theorem, string diagrams, pullback, pushforward 
@article{TAC_2014_29_a3,
     author = {Ryan Cohen Reich},
     title = {Obvious natural morphisms of sheaves are unique},
     journal = {Theory and applications of categories},
     pages = {48--99},
     publisher = {mathdoc},
     volume = {29},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2014_29_a3/}
}
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Ryan Cohen Reich. Obvious natural morphisms of sheaves are unique. Theory and applications of categories, Tome 29 (2014), pp. 48-99. http://geodesic.mathdoc.fr/item/TAC_2014_29_a3/