2022-01-22

Tags: #web/quick-notes

Refs: ?

Questions

#todo/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.

#web/quick-notes #todo/questions