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  "body": "\n## Summary\n\nA model category is a [[category|A2DE66937D]] with three chosen classes of morphisms, called weak equivalences, fibrations and cofibrations, that satisfy axioms modelled on topological spaces.\nQuillen introduced model categories in 1967 to do homotopy theory in settings such as simplicial sets and chain complexes.\nThe weak equivalences are the maps to be treated as isomorphisms, and the fibrations and cofibrations give the tools to compute with them.\nThe homotopy category of a model category, obtained by inverting the weak equivalences, has a concrete description in terms of homotopy classes of maps.\n\n## Explanations\n\n### L1 human\n\nIn topology one often wants to regard two spaces as the same when one can be deformed into the other.\nMaking such deformations into honest isomorphisms is easy to say but hard to compute with.\nA model category adds two kinds of well-behaved maps, fibrations and cofibrations, which play the roles of nice projections and nice inclusions.\nWith them one can replace any object by a good one and compute maps up to homotopy as in topology.\nTopological spaces, simplicial sets and chain complexes of modules all carry standard model structures.\n",
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