---
date: 2022-01-22
tags: [ web/quick-notes ]
---

# 2022-01-22

Tags:
#web/quick-notes

Refs:
?

## Questions

#todo/questions 

- What is a [conservative functor](conservative%20functor.md)?
- What is an [exit path category](exit%20path%20category)?
- What is a [Kan extension](Kan%20extension.md)?
- What is a [décollage](decollage)?
- What is a [semisimple category](semisimple%20category)?
- What is [Weil restriction](Weil%20restriction.md)?
- What is an [idemptotent-complete category](idemptotent-complete%20category.md)?

## 01:05

Complete Segal spaces:

![](figures/2022-01-22_01-11-28.png)

- Spaces can be [stratified](stratified.md) by posets.

- For $P$ a [poset](poset), $\mathrm{sd}(P)$ is the poset of linearly ordered finite subsets of $P$.

![](figures/2022-01-22_01-16-11.png)


## 01:20

Some [equivariant homotopy theory](equivariant%20homotopy%20theory.md).
See <https://arxiv.org/pdf/2010.15722.pdf>

![](figures/2022-01-22_01-20-05.png)

![](figures/2022-01-22_01-22-05.png)

- $\ho\Finset$ is the [Lawvere theory](Lawvere%20theory) for $\Comm\Mon$

![](figures/2022-01-22_01-34-27.png)

![](figures/2022-01-22_01-34-56.png)

![](figures/2022-01-22_01-37-16.png)

![](figures/2022-01-22_01-38-12.png)


- Representation theory over $\KU$ is a smooth deformation of representation theory of $\ZZpadic$.

- For $\cat C$ a category, the \(\K\dash\)theory space is $\Loop^\infty \K([\BG, \cat C])$ when $G$ is a finite group.
- Nice result: $\Rep(G)\slice \CC \cong \K_0 [\BG, \Vect^\fd \slice \CC]$.

![](figures/2022-01-22_01-43-53.png)

Commutative monoid objects in a category:

![](figures/2022-01-22_15-53-24.png)

[Polynomial functors](Polynomial%20functors)

![](figures/2022-01-22_15-55-43.png)

![](figures/2022-01-22_16-05-51.png)

![](figures/2022-01-22_16-06-17.png)

Analytic functors:

![](figures/2022-01-22_15-56-09.png)

Categorical $G\dash$spaces and orbits:

![](figures/2022-01-22_15-57-24.png)

$G\dash$symmetric monoid objects:

![](figures/2022-01-22_15-58-26.png)

Multiplications/folds and [transfers](transfers.md):

![](figures/2022-01-22_15-59-02.png)

Connection to Mackey functors:

![](figures/2022-01-22_15-59-16.png)


[Perfect objects](Perfect%20objects) (in spectral DM stacks):

![](figures/2022-01-22_16-04-37.png)

Algebraic \(\K\dash\)theory as a functor:

![](figures/2022-01-22_16-07-05.png)

Tambara functors:

![](figures/2022-01-22_16-09-18.png)

Genuine $(G, \EE_\infty)\dash$spectra:

![](figures/2022-01-22_16-09-49.png)

Algebraic \(\K\dash\)theory is a homotopical Tambara functor:

![](figures/2022-01-22_16-10-34.png)


For $\cat{C}$ a $G\dash$symmetric monoidal category in the naive sense, $\K(\cat C)$ is a [Green functor](Green%20functor).

![](figures/2022-01-22_16-11-23.png)

![](figures/2022-01-22_16-12-41.png)

## 16:50

[Derived categories](Derived%20categories): now the standard approach for [microlocal analysis](microlocal%20analysis), and a base for [noncommutative algebraic geometry](noncommutative%20algebraic%20geometry.md).
[Unsorted/HH](Unsorted/HH.md) can distinguish equivalence classes of derived categories.
Used in the representation theory of finite Chevalley groups.