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I have finally filled content into the entry derived functor in homological algebra.
That entry had existed in template form for years, with the intention to eventually take up that content, but clearly I had forgotten to actually put it there after I had written it out on my own web at HAI (schreiber). Now I have copied it over.
Nice work.
Still I am bit confused with the intended scope of the page. It starts talking about “total hyperderived functor” (I do not know what is hyper for here, but never mind) and it does start with mentioning a derived category, but the rest is about what are now called as you know the classical derived functors, that is, the derived functors in the setup of abelian categories a la Cartan-Eilenberg and Tohoku. Modern homological algebra by the (left or right) derived functor by default instead means the corresponding total derived functor between the appropriate derived categories, usually defined there by means of a Kan extension. Under mild conditions, they can be (of course) related to the classical derived functors.
That’s precisely the intention of the page, to go through the classical theory. I tried to say that at the beginning of the page, but if it remains unclear, please feel invited to add further commentary along these lines.
Then we have to change at some point the misleading title. E.g. to say not that this is derived functor in homological algebra but in Cartan-Eilenberg approach or whatever such. $n$Lab is considered supermodern so by homological algebra we should understand what is the modern usage of the subject by practitioners, and separate the “classical” 1950-s phase from 1960-s (Grothendieck-Verdier) and 1980-s on (enhanced derived categories). I do not think that the term classical refers to such homological algebra (as derived categories of 1960s are also classical) but that the specific term “classical derived functor” is established for the sequences of derived functors in the setup of abelian categories. The approach via satellites is also from that first period and is a variant of the same.
Okay, sure, please change the title to whatever seems to appropriately describe the content. Thanks.
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