# 2021-09-16

## 20:04

• 2-category : a category enriched in small categories, so hom sets are categories and compositions form bifunctors.

• Arrows in $$\mathsf{C}(x, y)(u\to v)$$ are deformations of $$u$$ to $$v$$.
• 2-functors are enriched functors.
• classifying space of a 2-cat: replace morphism cats $$C(x, y)$$ with $${\mathbf{B}}C(x, y)$$ to get a topological 1-cat, then take $${\mathbf{B}}C \coloneqq{ {\left\lvert {{ \mathcal{N}({C}) }} \right\rvert} }$$.

• For $$F:\mathsf{C}\to \mathsf{D}$$, fixing $$p\in D$$, can form a homotopy fiber 2-category $$y//F$$.

Then $${\mathbf{B}}F: {\mathbf{B}}C\to {\mathbf{B}}D$$ is a homotopy equivalence of spaces if $$B(y//F) \simeq{\operatorname{pt}}$$ is contractible for all $$y\in \mathsf{D}$$.

• homotopy fiber cat: $$y//F$$ is a lax comma category.

• lax functor :

• Any monoidal category is a 2-category with one object.
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