Tags: #higher-algebra/category-theory

  • 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.
#web/quick-notes #higher-algebra/category-theory