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With regards to model categories, what is the intuition behind the small object argument and cofibrantly generated model categories?

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    $\begingroup$ You want to produce some fibrant thing, say. So you need a bunch of diagrams to have lifts. So just make them have lifts in a stupid way by adjoining some stuff indexed over all "lifting problems". Whoops! There's too many lifting problems! So we hope that the class of such is generated by a set so that we don't break math. $\endgroup$ – Dylan Wilson Mar 1 '16 at 15:38

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