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:
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Spaces can be stratified by posets.
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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 \(\mathsf{Comm}\mathsf{Mon}\)
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Representation theory over \({\operatorname{KU}}\) is a smooth deformation of representation theory of \({ {\mathbf{Z}}_{\widehat{p}} }\).
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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.
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Nice result: \({\mathsf{Rep}}(G)_{/ {{\mathbf{C}}}} \cong {\mathsf{K}}_0 [{{\mathbf{B}}G}, { \mathsf{Vect} }^{\mathrm{fd}}_{/ {{\mathbf{C}}}} ]\).
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.