New York Journal of Mathematics
Volume 17 (2011) 75-125


Michael Shulman

Comparing composites of left and right derived functors

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Published: February 25, 2011
Keywords: Model category, derived functor, double category, adjunction, mate
Subject: 55U35 (Primary) and 18D05 (Secondary)

We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this double category. The theory of conjunctions and mates in double categories, which generalizes the theory of adjunctions and mates in 2-categories, then gives us canonical ways to compare composites of left and right derived functors. We give a number of sample applications, most of which are improvements of existing proofs in the literature.

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