Kerodon

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Definition 2.2.5.5. We let $\operatorname{2Cat}_{\operatorname{Lax}}$ denote the ordinary category whose objects are (small) bicategories and whose morphisms are lax functors between bicategories (Definition 2.2.4.5), with composition given by Construction 2.2.5.1 and identity morphisms given by Example 2.2.5.4. We define (non-full) subcategories

\[ \operatorname{2Cat}_{\operatorname{Str}} \subsetneq \operatorname{2Cat}\subsetneq \operatorname{2Cat}_{\operatorname{Lax}} \supsetneq \operatorname{2Cat}_{\operatorname{ULax}} \]

  • The objects of $\operatorname{2Cat}$ are bicategories, and the morphisms of $\operatorname{2Cat}$ are functors.

  • The objects of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict bicategories, and the morphisms of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict functors.

  • The objects of $\operatorname{2Cat}_{\operatorname{ULax}}$ are bicategories, and the morphisms of $\operatorname{2Cat}_{\operatorname{ULax}}$ are strictly unital lax functors.

We will refer to $\operatorname{2Cat}$ as the category of bicategories, and to $\operatorname{2Cat}_{\operatorname{Str}}$ as the category of strict bicategories.