2021-05-24

12:07

Reference: ???, GROOT

  • Big idea: free implies flat for algebras, is this true in the equivariant settings?

    • Almost all something are something, check talk title!!

  • Abelian groups \(\approx\) Mackey functor.

  • \({\mathbb{Z}}\approx\) Burnside Mackey functors

  • Commutative rings \(\approx\) Green functor \((E_\infty\) algebras), Incomplete functors, Tambara functor

  • Free algebra \({\mathbb{Z}}[G]\) comparable to free incomplete Tambara functor

  • Similarities come from being algebras over Operads.

  • Hill-Hopkins-Ravenel involves spectral sequences of Mackey functors

  • All rational Mackey functors are free

  • \(A^{\mathcal{O}}[x_{G/H}]\) is almost never flat.

  • A Mackey functor is an additive functor \(M: A^g\to {\mathsf{Ab}}\), where \(A^G\) is the Burnside category : finite \(G{\hbox{-}}\)sets, where morphisms \(A^G(X, Y)\) is the group completion wrt \(\coprod\) of finite \(G{\hbox{-}}\)sets, so spans.

    • Composition of spans is pullback.
    • Sends disjoint unions 
      \to
      direct sums
    • Every object is the disjoint union of orbits \(G/H\)

  • To define a Mackey functor \(F\), it suffices to give abelian groups \(F(G/H)\) for \(H\leq G\), restrictions \(\operatorname{res}^H_K\), and transfer map \({\mathrm{tr}}_K^H\) in the target.

  • Burnside Mackey functor : \(\underline{A}\).

    • Objects are \(K_0\) of finite groups under \(\coprod\), \(\operatorname{res}\) is the forgetful functor, \({\mathrm{tr}}_K^H([x]) = [H { \underset{\scriptscriptstyle {K} }{\times} } X]\).

  • Theorem (Lewis): the category of Mackey functors is abelian, and has a [[symmetric monoidal category|symmetric monoidal]] product \(\boxtimes\) with unit \(\underline{A}\).

  • A Green functor is a monoid for \(\boxtimes\), which is an [[E_n ring spectrum|E_infty algebra]] in Mackey functors.

    • A Mackey functor \(R\) where \(R(G/H)\) is a unital commutative ring and \(\operatorname{res}^H_K\) is a ring morphism.

  • An incomplete Tambara functor is an \(N_\infty\) algebra in Mackey functors

  • A Tambara functor is a Green functor with that data of a norm map \(\nm_K^H\), a multiplicative morphism.

    • \(\underline{A}\) has norms given by \({\mathsf{Set}}^K(A, B)\), \(K{\hbox{-}}\)equivariant set functions.

  • Indexing systems: valid suborderings on the poset lattice of subgroups

  • Theorem (Barnes-Roitzheim-?) For \(C_{pq}\), there are roughly a Catalan's number of valid indexing systems.

17:45

Tags: #idle_thoughts

  • Really cool idea I like from that talk: what is the probability density of objects in a category?

    In a precise sense, what proportion of objects are projective, flat, free, dualizable, indecomposable, simple, etc?

  • I think I really want analytic structure on a category! I’m reminded of results like Morse functions being generic in spaces of functions, or perturbing [[Hamiltonian|Hamiltonians]] in Floer Theory. We can cook up topologies to make these kinds of statements precise in the classical setting….how can we do it here?

    • I’ve been thinking about “integrating over a category” a lot, some way to extract “average information” about a category. Integration on moduli spaces is hard!

  • Look up simple normal crossings divisor.

21:36

#quick_notes #unanswered_questions #idle_thoughts