Timeline for Is there a way to “derive” a (non-exact) functor which preserves images?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 4 at 0:27 | comment | added | Denis T | According to whether you want your derived functor to be defined on the bounded, bounded above/below, or the whole unbounded derived category of $A$, you'll have to put more and more niceness properties on $A$; but besides additivity, $F$ can be anything. | |
| Feb 4 at 0:23 | comment | added | Denis T | Suppose you have an additive functor $F: A \to B$ between abelian categories, and derived categories of $A$ and $B$ exist. Let $q_A$ and $q_B$ be the quotient functors from homotopy categories of complexes to the derived categories of $A$ and $B$. Left/right derived functor (if it exists) is a left/right Kan extension of [pointwise application of $F$ on homotopy categories composed with $q_B$] along $q_A$. Exactness properties of the initial functor really do not matter for the existence, it depends on whether acyclic complexes in $A$ form a (co)localizing subcategory. | |
| Feb 3 at 21:28 | comment | added | Tim Campion | @DenisT What form of "usual derived functor" applies to a functor which is neither left nor right exact? | |
| Feb 3 at 21:04 | comment | added | Denis T | Usual derived functors do the job just as well, in my opinion; you just do not have that zeroth derived functor is isomorphic to the original one. Do you have any reason to use something else? | |
| Feb 2 at 19:50 | history | asked | Tim Campion | CC BY-SA 4.0 |