---
date: 2021-06-05
tags: [ web/quick-notes ]
---


# 2021-06-05

##  Talbot, Lyne Moser Part 1

- $\inftycatn{2}$: discrete set of objects, enriched in categories, $2\dash$morphisms are strictly associative?

- $\inftycatn{n}$: all $n+k\dash$morphisms are invertible.

- There is an embedding $n\dash\Cat \embeds \inftycatn{n}$with a specific model structure.

- Model structure on $\sSet$: fibrant objects are Kan complexes, there is a Quillen equivalence between $\Top$ and $\sSet$

- $\inftycatn{1}$: enriched in $\inftyGrpd$
  - [Quasicategory](Quasicategory.md) : lifting property with lifting against inner horns online

  ![](attachments/image_2021-06-05-12-16-29.png)

  - [Joyal model structure](Joyal%20model%20structure) : $\sSet$ with quasicategories as fibrant objects.

  - [Quillen equivalence](Quillen%20equivalence.md) to the [Kan model structure](Kan%20model%20structure) by taking the [homotopy coherent nerve](homotopy%20coherent%20nerve)?

- **Exercise**: [nerve](nerve.md) is a [Kan complex](Kan%20complex.md) iff $\cat{C}$ is a [groupoid](groupoid.md).

- Complete [Segal space](Segal%20space) : a [simplicial space](simplicial%20space) $W \Delta\op\to \sSet_{\Kan}$ with some conditions.

- [simplicial set](simplicial%20set.md) $\sSet^{\Delta\op}$ 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](cone%20category.md) :
  
  ![](attachments/image_2021-06-05-12-24-46.png)

  - Latter is more universal, only requires pullbacks and [cotensor](cotensor) to define?

- Terminal objects: isomorphisms from [slice category](slice%20category) to $\cat{C}$:

  ![](attachments/image_2021-06-05-12-25-36.png)

- Enriched definition of limits:

  ![](attachments/image_2021-06-05-12-27-23.png)

- Terminal objects: $X\slice{x} \mapsvia{\sim} X$ is a [weak equivalence](weak%20equivalence.md) in $\sSet_{\quasiCat}$.

- All definitions of limits in $\inftycatn{1}$ recover the usual notions via the nerve.

- **Exercise**: Show that the limit of $F: I\to \cat{C}$ is the limit of its [nerve](nerve.md)?

- **Theorem**: limit of $F$ in $\sSet_{\quasiCat}$ is a [homotopy limit](homotopy%20limit.md) of its [Unsorted/adjoint (categorical)](Unsorted/adjoint%20(categorical).md) in $\sSet\dash\Cat_{\Kan}$, and the limit of its adjoint in $\sSet^{\Delta\op}_\CSS$.

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

## 12:58

Things to look up from written notes:

- [Picard bundle](Picard%20bundle)
- [Hasse invariant](Hasse%20invariant.md)
- [Riemannian Geometry](Unsorted/Riemannian%20Geometry.md)
- [Frobenius lift](Frobenius%20lift)
- [lambda ring](lambda%20ring)
- [absolute Galois group](absolute%20Galois%20group.md)
- For [elliptic curves](elliptic%20curve.md) :
	- [level of an elliptic curve](level%20of%20an%20elliptic%20curve), [weight of an elliptic curve](weight%20of%20an%20elliptic%20curve), [conductor of an elliptic curve](conductor%20of%20an%20elliptic%20curve)
- What is the difference between [local class field theory](local%20class%20field%20theory) and [global class field theory](global%20class%20field%20theory)
- [theta function](theta%20function.md)

## Talbot, Lyne Moser Part 2

- Next: models of $\inftycat{\infty, 2}$

  - Can take [enrichment](enrichment) over $\quasiCat$ or $\CSS$

- [base change along a functor](base%20change%20along%20a%20functor)??

- [2-category](2-category.md) : categories [enriched](enriched) in categories

- Recall: $\CSS = \Fun(\Delta\op, \sSet)$.
	- We have $\Delta \subseteq \Cat$ a full subcategory, we now want a version for $2\dash\Cat$: $\Theta_2\op$.
	- Turns out to be [wreath product](wreath%20product.md) $\Delta\wreath\Delta$.
- Idea: keep track of 2-morphisms, i.e. two-cells, can keep all of their possible compositions

![](attachments/image_2021-06-05-13-09-28.png)

![](attachments/image_2021-06-05-13-09-54.png)

![](attachments/image_2021-06-05-13-10-05.png)