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.

• ${\mathbf{Z}}\approx$ Burnside Mackey functors

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

• Free algebra ${\mathbf{Z}}[G]$ 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.

  • Are transfers like inflation? #todo/questions

• 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 product $\boxtimes$ with unit $\underline{A}$.

• A Green functor is a monoid for $\boxtimes$, which is an 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 Unsorted/field 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: #personal/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 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

• The Gelfand representation is really cool. Look into how this duality shows up for schemes!

• What is the length of a module?