derived smooth geometry
What is known as (Grothendieck’s) six operations is a formalization of structure that
These are four operations and together with
form six operations.
(All this is usually interpreted as derived functors/(infinity,1)-functors so that for instance in the usual application to derived categories of abelian sheaves the last two operations are really Tor and Ext.)
With a list of compatibility conditions between these (for instance (Cisinski-Déglise 09, p. x) this is a structure of Grothendieck’s six operations.
These consistency conditions include the following:
The adjunctions are functorial, hence form 2-functors , :
Often specializations of the general concept play a role:
Verdier-Grothendieck context: the “projection formula” holds naturally in
Grothendieck context: and the projection formula holds .
The initial object in the (infinity,2)-category of functors to stable (infinity,1)-categories which satisfy the six operations formalism (and a bit more, such that A1-homotopy invariance) is stable motivic homotopy theory. See there for more.
General abstract discussion is in
The traditional applications are discussed in
Joseph Lipman, Notes on derived functors and Grothendieck duality, in Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Math., no. 1960, Springer-Verlag, New York, 2009, 1–259. (pdf)
Yves Laszlo, Martin Olsson, The six operations for sheaves on Artin stacks I: Finite Coefficients (arXiv:math/0512097)
A quick list of the axioms with a Grothendieck’s six operations with an eye towards the definition of motives is in section A.5 of