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  "body": "\n## Definition\n\nA *bicategory* $\\mathcal{B}$ consists of:\n\n1. a class of objects;\n2. for each pair of objects $a, b$, a [[category|A2DE66937D]] $\\mathcal{B}(a, b)$ of 1-morphisms $f \\colon a \\to b$ and 2-morphisms $\\theta \\colon f \\Rightarrow g$;\n3. for each triple $a, b, c$, a [[functor|28NQ0KRHD3]] $\\circ \\colon \\mathcal{B}(b, c) \\times \\mathcal{B}(a, b) \\to \\mathcal{B}(a, c)$, called horizontal composition;\n4. for each object $a$, an identity 1-morphism $1_a \\in \\mathcal{B}(a, a)$;\n5. invertible 2-morphisms $\\alpha_{h,g,f} \\colon (h \\circ g) \\circ f \\Rightarrow h \\circ (g \\circ f)$, $\\lambda_f \\colon 1_b \\circ f \\Rightarrow f$ and $\\rho_f \\colon f \\circ 1_a \\Rightarrow f$, [[natural|KYCPCC57FT]] in $f$, $g$, $h$.\n\nThese data satisfy, for all composable $f \\colon a \\to b$, $g \\colon b \\to c$, $h \\colon c \\to d$, $k \\colon d \\to e$:\n\n1. pentagon: $(1_k \\circ \\alpha_{h,g,f}) \\cdot \\alpha_{k, h \\circ g, f} \\cdot (\\alpha_{k,h,g} \\circ 1_f) = \\alpha_{k,h,g \\circ f} \\cdot \\alpha_{k \\circ h, g, f}$;\n2. triangle: $(1_g \\circ \\lambda_f) \\cdot \\alpha_{g, 1_b, f} = \\rho_g \\circ 1_f$.\n\nHere $\\cdot$ is vertical composition in the hom-categories, and $1_f$ also denotes the identity 2-morphism of $f$.\n\n## Remarks\n\nExample: the bicategory $\\mathbf{Bim}$ has rings as objects, $S$-$R$-bimodules as 1-morphisms $R \\to S$, and bimodule maps as 2-morphisms.\nIts horizontal composition is $N \\circ M = N \\otimes_S M$, its identity is $1_R = R$, and $\\alpha$, $\\lambda$, $\\rho$ are the usual isomorphisms of tensor products.\nThe pentagon and triangle hold because, on elementary tensors, both sides of each identity give the same rebracketed tensor.\nNon-example: in the same data, replace each identity $1_R$ by the zero bimodule; then $0 \\otimes_R M = 0$, which is not isomorphic to a nonzero $M$, so no unitor $\\lambda_M$ exists.\nA one-object bicategory is the same as a [[monoidal category|M3MSQ3NKAE]], and a bicategory whose $\\alpha$, $\\lambda$, $\\rho$ are identities is a [[2-category|CQACXG10JZ]].\n",
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