Daily six-functor

I will prove the commutativity of some diagram every day. Started from May 1st, 2024.

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Plan

• Compatibility of exchange maps among 3 of six-functors: There are 56 diagrams.

• Compatibility between six functors and one of the following transforms:

 *-unit/counit, !-unit/counit, tensor-unit/counit, projection formula, forgetting support map, immersion transform, purity transform, relative *-adjunction , Poincare-Verdier duality.

There are 72 diagrams.

• Compatibility among two of the above 12 transforms: 66 diagrams to check.

• Special day: Coherence theorems. 

Will show that we can reduce the compatibility of exchange maps among n of six-functors to the case when n = 3.

• Verdier duality v.s. everything: There are 33 diagrams (though all of them are immediate consequence of already proved statements).

• Generalization to pushable/pullable squares.

In the future, I plan to prove the commutativity of diagrams involving following functors/ transforms as well:

nearby/vanishing cycle functor (via Koszul duality), specialization, Fourier-Wang transform, microlocalization, Milnor sequence, virtual pullback, contraction transforms, Braden's transform for hyperbolic localizations, ...