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  "body": "\n## Summary\n\nAn $(\\infty,1)$-category is a higher category with $k$-morphisms for every $k \\ge 1$, in which all $k$-morphisms with $k \\ge 2$ are invertible.\nIt behaves like a [[category|A2DE66937D]] whose hom-sets are replaced by spaces, taken up to homotopy.\nSeveral precise models exist, such as [[quasi-categories|9328WEVJEK]], categories enriched in Kan complexes, and complete Segal spaces, and these models are known to be equivalent.\n$(\\infty,1)$-categories are the natural setting for homotopy theory, derived algebraic geometry and stable homotopy theory.\n\n## Explanations\n\n### L1 human\n\nIn topology, two maps can be homotopic, two homotopies can be homotopic, and so on.\nAn $(\\infty,1)$-category keeps all of this information instead of collapsing homotopic maps to a single class.\nThe \"$1$\" means that only the ordinary morphisms may fail to be invertible; everything above them is a kind of homotopy, which can be reversed.\nFor example, topological spaces with continuous maps, homotopies between maps and homotopies between homotopies form an $(\\infty,1)$-category.\n",
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  "url": "https://tamnd.github.io/math2/wiki/1-category/",
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