Comparing composites of left and right derived functors

By Michael Shulman. 2007

62 citations recorded by the source corpus.

We introduce a new categorical framework for studying de- rived 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 ar- rows 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 canon- ical 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/258d7c0bcc4df530afb7af7a7d97e65e2fb33bcc

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In Algebra and Number Theory

1. Michael Shulman
2. Emily Riehl
3. R. Haugseng

In Geometry and Topology

1. Michael Shulman
2. Dominic R. Verity
3. Emily Riehl
4. R. Haugseng

In Mathematical Physics

1. Michael Shulman
2. Dominic R. Verity
3. Denis-Charles Cisinski
4. Emily Riehl
5. R. Haugseng

Citations in the corpus

• HOMOTOPY LIMITS AND COLIMITS AND ENRICHED HOMOTOPY THEORY (2006, background): “general, left adjoints will have left derived functors, while right adjoints will have right derived functors, but we have no more general theory explaining why this should be true. However, see also [Shu07]. 5. Homotopy colimits as derived functors We now begin the second part of the paper and desc”