---
date: 2021-05-24
tags: [ web/quick-notes ]
---


# 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](Mackey%20functor.md).

- $\ZZ \approx$ Burnside Mackey functors

- Commutative rings $\approx$ [Green functor](Green%20functor) $(E_\infty$ algebras), Incomplete functors, [Tambara functor](Tambara%20functor)

- Free algebra $\ZZ[G]$ comparable to free incomplete Tambara functor

- Similarities come from being algebras over [Operads](Operads).

- [Hill-Hopkins-Ravenel](Hill-Hopkins-Ravenel.md) involves spectral sequences of Mackey functors

- All rational Mackey functors are free

- $A^\OO [x_{G/H}]$ is almost never flat.

- A Mackey functor is an additive functor $M: A^g\to \Ab$, where $A^G$ is the [Burnside category](Burnside%20category) : finite $G\dash$sets, where morphisms $A^G(X, Y)$ is the group completion wrt $\coprod$ of finite $G\dash$sets, so [spans](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 $\res^H_K$, and [transfer map](transfer%20map) $\tr_K^H$ in the target.

	- Are transfers like inflation? #todo/questions 

- [Burnside Mackey functor](Burnside%20Mackey%20functor) : $\ul A$. 
	- Objects are $K_0$ of finite groups under $\coprod$, $\res$ is the forgetful functor, $\tr_K^H([x]) = [H \fiberprod{K} X]$.

- **Theorem** (Lewis): the category of Mackey functors is abelian, and has a [symmetric monoidal](symmetric%20monoidal%20category) product $\boxtimes$ with unit $\ul A$.

- A [Green functor](Green%20functor) is a monoid for $\boxtimes$, which is an [E_infty algebra](E_n%20ring%20spectrum.md) in Mackey functors.
	- A Mackey functor $R$ where $R(G/H)$ is a unital commutative ring and $\res^H_K$ is a ring morphism.

- An *incomplete Tambara functor* is an $N_\infty$ algebra in Mackey functors

- A [Tambara functor](Tambara%20functor) is a [Green functor](Green%20functor) with that data of a [Unsorted/field norm](Unsorted/field%20norm.md) map $\nm_K^H$, a multiplicative morphism.

	- $\ul A$ has norms given by $\Set^K(A, B)$, $K\dash$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](Catalan's%20number) 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](Morse%20functions) being generic in spaces of functions, or perturbing [Hamiltonians](Hamiltonian.md) in [Floer Theory](Floer%20Theory.md).
  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](simple%20normal%20crossings.md) divisor.

## 21:36

- The [Gelfand representation](Gelfand%20representation.md) is really cool.
	Look into how this duality shows up for schemes!

- What is the [length of a module](length%20of%20a%20module.md)?