I belive we study them because in important categories they are close to free objects and even a retract of a free object in some algebraic instances (for example, direct summands in Mod_R, and precisely the free objects in Gps). Am I right? Are there other reasons?
Why are projective objects important?
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abstract-algebra
group-theory
category-theory
homological-algebra
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10[For deriving functors](http://en.wikipedia.org/wiki/Derived_functor). – 2011-07-16
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2Seconding Theo B.'s comment: while one can define "universal delta-functor" without reference to any particular class of "resolutions" for _computing_, very quickly the _mapping_ properties of projective/injectives (which yield chain homotopies proving independence of choice... etc.) become compellingly interesting for both computation and to prove _existence_ of the universal delta-functors. These are the very simplest _acyclic_ objects, being universally so (whenever proj/inj make sense). Even if we declare them somewhat auxiliary, they play a pivotal role. – 2011-07-16
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0I think your question is a little ambiguous: in an arbitrary category (and even abelian category) projective and injective objects play a perfectly dual role (i.e., upon switching to the opposite category). Thus the right answer to give in this level of generality is universal $\delta$-functors. – 2011-07-16
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0But on the other hand, in the category of modules over a ring, projective and injective objects behave very differently: e.g. free modules are necessarily projective but rarely injective. And indeed projective **modules** are interesting for lots of things other than the machinery of homological algebra. So which are you interested in, arbitrary categories or modules over a ring? – 2011-07-16