2021-06-05

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
  attachments/image_2021-06-05-12-16-29.png❗

  • 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 :

  attachments/image_2021-06-05-12-24-46.png❗
  • Latter is more universal, only requires pullbacks and cotensor to define?

• Terminal objects: isomorphisms from slice category to \(\mathsf{C}\):

  attachments/image_2021-06-05-12-25-36.png❗

• Enriched definition of limits:

  attachments/image_2021-06-05-12-27-23.png❗

• 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 adjoint 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.

12:58

Things to look up from written notes:

• Picard bundle
• Hasse invariant
• Connection on a bundle
• 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