---
date: 2021-09-16
tags: [ web/quick-notes ]
---


# 2021-09-16

## 20:04

Tags:
#higher-algebra/category-theory 

- [2-category](2-category.md) : a category [enriched](Unsorted/enriched%20category.md) in small categories, so hom sets are categories and compositions form bifunctors.
  - Arrows in $\cat C(x, y)(u\to v)$ are [deformations](deformations) of $u$ to $v$. 
  - 2-functors are enriched functors.

- [classifying space](classifying%20space.md) of a 2-cat: replace morphism cats $C(x, y)$ with $\B C(x, y)$ to get a topological 1-cat, then take $\B C \da \realize{\nerve{C}}$.

- For $F:\cat C\to \cat D$, fixing $p\in D$, can form a [homotopy fiber](homotopy%20fiber.md) 2-category $y//F$.

  Then $\B F: \B C\to \B D$ is a homotopy equivalence of spaces if $B(y//F) \homotopic \pt$ is contractible for all $y\in \cat D$.

- [homotopy fiber](homotopy%20fiber.md) cat: $y//F$ is a lax [comma category](comma%20category).

- [lax functor](lax%20functor.md) :

![](attachments/2021-09-16_20-13-11.png)

- Any [monoidal category](monoidal%20category.md) is a [2-category](2-category.md) with one object.