Talbot, Lyne Moser Part 1

  • \({ \underset{(\infty, {2})}{ \mathsf{Cat}} }\): discrete set of objects, enriched in categories, \(2{\hbox{-}}\)morphisms are strictly associative?

  • \({ \underset{(\infty, {n})}{ \mathsf{Cat}} }\): all \(n+k{\hbox{-}}\)morphisms are invertible.

  • There is an embedding \(n{\hbox{-}}\mathsf{Cat}\hookrightarrow{ \underset{(\infty, {n})}{ \mathsf{Cat}} }\)with a specific model structure.

  • Model structure on \({\mathsf{sSet}}\): fibrant objects are Kan complexes, there is a Quillen equivalence between \({\mathsf{Top}}\) and \({\mathsf{sSet}}\)

  • \({ \underset{(\infty, {1})}{ \mathsf{Cat}} }\): enriched in \({ \underset{\infty}{ {\mathsf{Grpd}}} }\)

    • Quasicategory : lifting property with lifting against inner horns online


    • Joyal model structure : \({\mathsf{sSet}}\) with quasicategories as fibrant objects.

    • Quillen equivalence to the Kan model structure by taking the homotopy coherent nerve?

  • Exercise: nerve is a Kan complex iff \(\mathsf{C}\) is a groupoid.

  • Complete Segal space : a simplicial space \(W \Delta^{\operatorname{op}}\to {\mathsf{sSet}}_{{\mathsf{Kan}}}\) with some conditions.

  • simplicial set \({\mathsf{sSet}}^{\Delta^{\operatorname{op}}}\) has a model structure where complete Segal spaces are the fibrant objects

  • Exercise: if \(W\) is a complete Segal space then \(i_1^* W\) is a quasicategory where \(i_1: \Delta \to \Delta^{\times 2}\) sends \([n]\) to \(([n], [0])\).

  • Limits: isomorphisms on hom sets, or terminal objects in the cone category :


    • Latter is more universal, only requires pullbacks and cotensor to define?
  • Terminal objects: isomorphisms from slice category to \(\mathsf{C}\):


  • Enriched definition of limits:


  • Terminal objects: \(X_{/ {x}} \xrightarrow{\sim} X\) is a weak equivalence in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\).

  • All definitions of limits in \({ \underset{(\infty, {1})}{ \mathsf{Cat}} }\) recover the usual notions via the nerve.

  • Exercise: Show that the limit of \(F: I\to \mathsf{C}\) is the limit of its nerve?

  • Theorem: limit of \(F\) in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\) is a homotopy limit of its Unsorted/adjoint (categorical) in \({\mathsf{sSet}}{\hbox{-}}\mathsf{Cat}_{{\mathsf{Kan}}}\), and the limit of its adjoint in ${\mathsf{sSet}}^{\Delta^{\operatorname{op}}}_{ \mathsf{CSS} } $.

  • Upshot: many different models, can move between different models.


Things to look up from written notes:

  • Picard bundle
  • Hasse invariant
  • connection
  • Frobenius lift
  • lambda ring
  • absolute Galois group
  • For elliptic curves :
    • level of an elliptic curve, weight of an elliptic curve, conductor of an elliptic curve
  • What is the difference between local class field theory and global class field theory
  • theta function

Talbot, Lyne Moser Part 2

  • Next: models of \({ \underset{\infty}{ \mathsf{Cat}} }{\infty, 2}\)

    • Can take enrichment over ${ \mathsf{quasiCat} } $ or ${ \mathsf{CSS} } $
  • base change along a functor??

  • 2-category : categories enriched in categories

  • Recall: \({ \mathsf{CSS} } = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{sSet}})\).

    • We have \(\Delta \subseteq \mathsf{Cat}\) a full subcategory, we now want a version for \(2{\hbox{-}}\mathsf{Cat}\): \(\Theta_2^{\operatorname{op}}\).
    • Turns out to be wreath product \(\Delta\wr\Delta\).
  • Idea: keep track of 2-morphisms, i.e. two-cells, can keep all of their possible compositions




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