Assume that a functor on the category of groups vanishes on all projective objects. Is it necessarily the left derived functor of a half exact functor on this category?
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3$\begingroup$ Is this true for functors out of abelian categories? I wasn't aware. $\endgroup$– Mark GrantCommented Aug 31, 2023 at 9:11
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4$\begingroup$ do you mean an endofunctor? or you mean "to itself"? In any case, are there derived functors in such a non-additive setting? $\endgroup$– YCorCommented Aug 31, 2023 at 9:15
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3$\begingroup$ What is the model structure on the category of groups? Or is it just abelian groups?? $\endgroup$– Bugs BunnyCommented Aug 31, 2023 at 15:28
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1$\begingroup$ @AliTaghavi: My comment was really about the motivation for the question. If you don't know this to be true in abelian categories, why would you suspect it to be true in the (non-abelian) category of groups? $\endgroup$– Mark GrantCommented Aug 31, 2023 at 19:43
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3$\begingroup$ By the way, there exists a subject of non-abelian homological algebra (see work of Barr-Beck, Inassaridze and others). The projective objects in the category of groups are precisely the free groups. $\endgroup$– Mark GrantCommented Aug 31, 2023 at 19:44
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