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  "body": "\n## Definition\n\nA *2-category* (or strict 2-category) is a category [[enriched|BPH9SJ30Y0]] over the [[category of small categories|ZY6R4H65P8]] $\\mathbf{Cat}$ with the Cartesian product as tensor product.\nExplicitly, a 2-category $\\mathcal{K}$ consists of:\n\n1. a class of objects;\n2. for each pair of objects $a, b$, a [[category|A2DE66937D]] $\\mathcal{K}(a, b)$ of 1-morphisms $f \\colon a \\to b$ and 2-morphisms $\\theta \\colon f \\Rightarrow g$, whose composition is called vertical;\n3. for each triple $a, b, c$, a [[functor|28NQ0KRHD3]] $\\circ \\colon \\mathcal{K}(b, c) \\times \\mathcal{K}(a, b) \\to \\mathcal{K}(a, c)$, called horizontal composition;\n4. for each object $a$, an identity 1-morphism $1_a$;\n\nsuch that horizontal composition is strictly associative and $1_a$ and its identity 2-morphism are strict units for it.\n\n## Remarks\n\nThat horizontal composition is a functor is the interchange law: $(\\psi' \\cdot \\psi) \\circ (\\theta' \\cdot \\theta) = (\\psi' \\circ \\theta') \\cdot (\\psi \\circ \\theta)$, where $\\cdot$ is vertical composition.\nExample: $\\mathbf{Cat}$, with small categories as objects, functors as 1-morphisms and [[natural transformations|KYCPCC57FT]] as 2-morphisms, is a 2-category; composition of functors is strictly associative.\nNon-example: rings, bimodules and bimodule maps, with composition of bimodules given by the tensor product, do not form a 2-category.\nTake an $R$-$S$-bimodule $M$, an $S$-$T$-bimodule $N$ and a $T$-$U$-bimodule $P$.\nThe bimodules $(M \\otimes_S N) \\otimes_T P$ and $M \\otimes_S (N \\otimes_T P)$ are isomorphic but not equal in general, so strict associativity fails; they form a bicategory instead.\n",
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