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This is a very imprecise, general question. Mostly because I'm not exactly sure what I'm after. I just think that I miss something crucial here.

In the context of model categories, homotopy theory, derived geometry and operads ect., it is often a major step to replace a structure, an object or something with a (co)fibrant replacement.

What do we achieve by this? Why is it "better" to work with these replacements, once we add any notion of "homotopy" into our theory? (I know there is no precise meaning of the word "better" in this context, I just don't see the reason, why people put so much thought into these replacements).

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    $\begingroup$ It might help to look at the definition. They are equivalent objects that have nice lifting properties. You probably have to look at a variety of examples to understand why nice lifting properties are nice. $\endgroup$ – Justin Young Feb 1 '18 at 17:55
  • $\begingroup$ Ok, it might boil down to the lifting properties. Thats a valueable starting point. Do you have particular interesting examples to offer, why nice lifting properties are nice? $\endgroup$ – Mark Neuhaus Feb 1 '18 at 18:43
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    $\begingroup$ Cofibrant replacements are better because left Quillen functors preserve weak equivalences between cofibrant objects, but not weak equivalences in general. Thus cofibrant replacements can be used to derive left Quillen functor, i.e., extract an ∞-functor from an ordinary functor. $\endgroup$ – Dmitri Pavlov Feb 2 '18 at 19:59
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    $\begingroup$ It is good to have a sizable collection of examples of such (co)fibrant object before we knew they were part of such structure. After all, the analysis of why such choices were so successful led, in part, to the crystallization of model categories and the like. Hovey's book has plenty of examples, and certainly reading about rational homotopy theory (and Quillen's original work) should prove helpful. $\endgroup$ – Pedro Tamaroff Feb 3 '18 at 0:39
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    $\begingroup$ (I somehow have the image of cofibrant replacements ---for example, projective resolutions of modules, complexes and DG-modules, minimal models, and the like--- to be 'fattenings' of our object, that replace them by some object that is more organized and unwraps the original object in a manageable way. The weak equivalence somehow comes from putting this object back together and collapsing it to the messy form it had before. This vague idea is made precise, for example, when one computes Hochschild cohomology and has to resolve an algebra.) $\endgroup$ – Pedro Tamaroff Feb 3 '18 at 0:47
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Cofibrant replacements are better because left Quillen functors preserve weak equivalences between cofibrant objects, but not weak equivalences in general. Thus cofibrant replacements can be used to derive left Quillen functor, i.e., extract an ∞-functor from an ordinary functor.

See Dwyer–Hirschhorn–Kan–Smith's “Homotopy limit functors on model categories and homotopical categories” for an overview of the modern theory of derived functors.

Relative categories and relative functors form one model for (∞,1)-categories. Specifically, they form a model category that is Quillen equivalent to other model categories: quasicategories (i.e., simplicial sets with the Joyal model structure) and Rezk's complete Segal spaces. This is shown in Barwick–Kan's “Relative categories: another model for the homotopy theory of homotopy theories” and Joyal–Tierney's “Quasi-categories vs Segal spaces”.

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