#
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.

http://www.tac.mta.ca/tac/volumes/29/4/29-04.pdf

TAC Home