Theory and applications of categories · 2003 · 60 citations · 9 references
In many areas of algebraic geometry, topology, and homological algebra a functor can have both left and right adjoints, with the right adjoint isomorphic to a shift of the left adjoint via a dualizing object, thereby rendering the more mysterious right adjoint into familiar terms. The paper offers a categorical examination of these results, distinguishing between the classical algebraic‑geometric framework and a distinct algebraic‑topological one, and clarifying which proof components are purely formal. This analysis streamlines the proofs of specific instances, as demonstrated in a subsequent work applying the theory to equivariant stable homotopy theory.
There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right ad- joint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate dualizing object. Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in fa- miliar terms. We give a categorical discussion of such results. One essential point is to differentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely different, framework that arises in algebraic topol- ogy. Another is to make clear which parts of the proofs of such results are formal. The analysis significantly simplifies the proofs of particular cases, as we illustrate in a sequel discussing applications to equivariant stable homotopy theory.
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Edgar H. Brown · Transactions of the American Mathematical Society · 1965 · 91 citations · Full text