# 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

• 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.

## 12:58

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