Bicategories

Definition

Def Bicategory A bicategory or weak $2$-category is a tuple $\text{require}mhchem(\mathsf{B},1,c,a,l,r)$ consisting of

• A category $\mathsf{B}$;
• Hom Categories: for each pair of objects ($0$-cells) $X,Y\in \text{obj}\mathsf{B}={\mathsf{B}}_0$, $\mathsf{B}$ is equipped with a category $\mathsf{B}(X,Y)$, called a hom category.
  • Its objects are called $1$-cells in $\mathsf{B}$. The collection of all the $1$-cells ($1$-morphisms) in $\mathsf{B}$ is denoted by ${\mathsf{B}}_1$.
  • Its morphisms are called $2$-cells in $\mathsf{B}$. The collection of all the $2$-cells in $\mathsf{B}$ is denoted by ${\mathsf{B}}_2$.
  • Composition law and identity morphisms in the category $\mathsf{B}(X,Y)$ are called vertical composition and identity $2$-cells, respectively.
  • An isomorphism in $\mathsf{B}(X,Y)$ is called an invertible $2$-cell, and its inverse is called a vertical inverse. For a $1$-cell $f$, its identity $2$-cell is denoted by $1_f$.
• Identity $1$-Cells: For each object $X\in {\mathsf{B}}_0$, denote $1$ as the singleton category, $1_X:1\to \mathsf{B}(X,X)$is a functor. We identify the functor $1_X$ with the $1$-cell $1_X(\ast )\in \mathsf{B}(X,X)$, called the identity $1$-cell of $X$.
• Horizontal Composition: For each triple of objects $X,Y,Z\in {\mathsf{B}}_0$, $c_{X,Y,Z}:\mathsf{B}(Y,Z)\times \mathsf{B}(X,Y)\to \mathsf{B}(X,Z)$is a functor, called the horizontal composition. For $1$-cells $f\in \mathsf{B}(X,Y{)}_0$ and $g\in \mathsf{B}(Y,Z{)}_0$, and $2$-cells $\alpha \in B(X,Y{)}_1$ and $\beta \in B(Y,Z{)}_1$, we use the notations $c_{X,Y,Z}(g,f)=g\circ f\text{ or }gf,c_{X,Y,Z}(\alpha ,\beta )=\alpha \ast \beta$
• Associator: For objects $W,X,Y,Z\in {\mathsf{B}}_0$, $a_{W,X,Y,Z}:c_{W,X,Z}\left(c_{X,Y,Z}\times {\mathrm{Id}}_{\mathsf{B}(W,X)}\right)\to c_{W,Y,Z}\left({\mathrm{Id}}_{\mathsf{B}(Y,Z)}\times c_{W,X,Y}\right)$is a natural isomorphism, called the associator, between functors $\mathsf{B}(Y,Z)\times \mathsf{B}(X,Y)\times \mathsf{B}(W,X)\to \mathsf{B}(W,Z)$
• Unitors: For each pair of objects $X,Y\in {\mathsf{B}}_0$, $c_{XYY}(1_Y\times {\mathrm{Id}}_{\mathsf{B}(X,Y)})\stackrel{{\ell }_{XY}}{\to }{\mathrm{Id}}_{\mathsf{B}(X,Y)}\stackrel{r_{XY}}{\leftarrow }{c}_{XXY}({\mathrm{Id}}_{\mathsf{B}(X,Y)}\times 1_X)$are natural isomorphisms, called the left unitor and the right unitor, respectively.

And the following axioms hold for $1$-cells $f\in \mathsf{B}(V,W{)}_0$, $g\in \mathsf{B}(W,X{)}_0$, $h\in \mathsf{B}(X,Y{)}_0$ and $k\in \mathsf{B}(Y,Z{)}_0$:

• Unity Axiom: The middle unity diagram commutes:
• Pentagon Axiom:

We usually abbreviate a bicategory as above to $\mathsf{B}$.

Definition

Def Local Property Suppose $P$ is a property of categories. A bicategory $\mathsf{B}$ is locally $P$ if each hom category in $\mathsf{B}$ has property $P$. In particular, $\mathsf{B}$ is:

• locally small if each hom category is a small category.
• locally discrete if each hom category is discrete.
• locally partially ordered if each hom category is a partially ordered set regarded as a small category.