# 2022-01-22

Tags: #web/quick-notes

Refs: ?

## Questions

• What is a conservative functor?
• What is an exit path category?
• What is a Kan extension?
• What is a décollage?
• What is a semisimple category?
• What is Weil restriction?
• What is an idemptotent-complete category?

## 01:05

Complete Segal spaces:

• Spaces can be stratified by posets.

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

## 01:20

Some equivariant homotopy theory. See https://arxiv.org/pdf/2010.15722.pdf

• $${\mathsf{ho}}{\mathsf{FinSet}}$$ is the Lawvere theory for $$\mathsf{Comm}\mathsf{Mon}$$

• Representation theory over $${\operatorname{KU}}$$ is a smooth deformation of representation theory of $${ {\mathbb{Z}}_{\widehat{p}} }$$.

• For $$\mathsf{C}$$ a category, the $${\mathsf{K}}{\hbox{-}}$$theory space is $${\Omega}^\infty {\mathsf{K}}([{{\mathbf{B}}G}, \mathsf{C}])$$ when $$G$$ is a finite group.

• Nice result: $${\mathsf{Rep}}(G)_{/ {{\mathbb{C}}}} \cong {\mathsf{K}}_0 [{{\mathbf{B}}G}, { \mathsf{Vect} }^{\mathrm{fd}}_{/ {{\mathbb{C}}}} ]$$.

Commutative monoid objects in a category:

Polynomial functors

Analytic functors:

Categorical $$G{\hbox{-}}$$spaces and orbits:

$$G{\hbox{-}}$$symmetric monoid objects:

Multiplications/folds and transfers:

Connection to Mackey functors:

Perfect objects (in spectral DM stacks):

Algebraic $${\mathsf{K}}{\hbox{-}}$$theory as a functor:

Tambara functors:

Genuine $$(G, {\mathbb{E}}_\infty){\hbox{-}}$$spectra:

Algebraic $${\mathsf{K}}{\hbox{-}}$$theory is a homotopical Tambara functor:

For $$\mathsf{C}$$ a $$G{\hbox{-}}$$symmetric monoidal category in the naive sense, $${\mathsf{K}}(\mathsf{C})$$ is a Green functor.

## 16:50

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

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