Isomorphisms between left and right adjoints

By H. Fausk, P. Hu, J. May. Theory and Applications of Categories, 2002

80 citations recorded by the source corpus.

There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, 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 familiar 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 topology. 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/03ee9f61bad75cdfb9eb90003174672dd7c88c78

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Citations in the corpus

Nilpotence and descent in equivariant stable homotopy theory (2015, background): “Once again, the map arises from universal properties, as explained in [27].”
Comparing composites of left and right derived functors (2007, methodology): “is. This alternate condition is sometimes easier to verify. There are many other mates of this sort that compare various composites of adjoint functors between monoidal categories; see, for instance, [FHM03]. The thing to notice about both of these examples is that the given structure uniquely speci”
Parametrized homotopy theory (2004, methodology): “A general categorical study of isomorphisms between left and right adjoints was given by Fausk, Hu, and May in [62], and a simplified proof of the Wirthmüller isomorphism from that perspective was given in [125].”
Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map (2011, background): “n there are natural equivalences: (1) f∗(Y ⊗T Z) ≃f∗Y ⊗S f∗Z (2) FT (Y,f∗X) ≃f∗FS(f∗Y,X), (3) f∗FT (Y,Z) ≃FS(f∗Y,f∗Z), (4) f!(f∗Y ⊗S X) ≃Y ⊗T f!X, (5) FT (f!X,Y) ≃f∗FS(X,f∗Y ). Proof. As explained in [27], we can deduce all of these equivalences from (1) and the projection formula (4). First, the eq”
Grothendieck–Neeman duality and the Wirthmüller isomorphism (2015, methodology): “Here, we sometimes add the adjective “strong” to avoid collision with Fausk-Hu-May’s slightly different notion of “Wirthmüller context” [FHM03]; see more in Remark 4.3.”