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  "body": "\n## Definition\n\nAn *$(\\infty,1)$-category* is a higher category with $k$-morphisms for every $k \\ge 1$, in which every $k$-morphism with $k \\ge 2$ is invertible.\nThis informal notion is made precise by a choice of model; the most used one is the following.\nIn the quasi-categorical model, an $(\\infty,1)$-category is a [[quasi-category|9328WEVJEK]]: a [[simplicial set|WW81ARB8K9]] in which every inner horn $\\Lambda^n_k \\to X$, $0 \u003c k \u003c n$, extends to the $n$-simplex $\\Delta^n$.\n\n## Remarks\n\nOther standard models are:\n\n1. simplicial categories, that is, categories [[enriched|BPH9SJ30Y0]] over simplicial sets, whose hom-objects are all [[Kan complexes|FDFYCM3YYC]];\n2. complete Segal spaces, which are certain simplicial objects in simplicial sets;\n3. Segal categories.\n\nEach model carries a [[model category|SDTZVZ6B4Q]] structure, and these model categories are connected by Quillen equivalences, so the models give the same homotopy theory of $(\\infty,1)$-categories.\nExample: the nerve of an ordinary category is a quasi-category, so every category is an $(\\infty,1)$-category with only identity higher morphisms.\nExample: Kan complexes are the $(\\infty,1)$-categories in which every morphism is invertible, the $\\infty$-groupoids.\nNon-example: $\\mathbf{Cat}$, viewed as a strict $2$-category, is not an $(\\infty,1)$-category in its own right.\nIt has non-invertible natural transformations as $2$-morphisms, while the $2$-morphisms of an $(\\infty,1)$-category must be invertible.\n",
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