Comparing composites of left and right derived functors

Details Comparing composites of left and right derived functors Comparing composites of left and right derived functors

2007-06-19

arxiv.org/abs/0706.2868v2

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

Mathematics

Category Theory

44 pages; expanded with detailed examples added; final version published in NYJM

Scientific paper

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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