Homology is an additive functor #
When V
is preadditive, HomologicalComplex V c
is also preadditive,
and homologyFunctor
is additive.
TODO: similarly for R
-linear.
The i
-th component of a chain map, as an additive map from chain maps to morphisms.
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An additive functor induces a functor between homological complexes. This is sometimes called the "prolongation".
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The functor on homological complexes induced by the identity functor is isomorphic to the identity functor.
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A natural transformation between functors induces a natural transformation between those functors applied to homological complexes.
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A natural isomorphism between functors induces a natural isomorphism between those functors applied to homological complexes.
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An equivalence of categories induces an equivalences between the respective categories of homological complex.
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Turning an object into a complex supported at j
then applying a functor is
the same as applying the functor then forming the complex.
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Turning an object into a chain complex supported at zero then applying a functor is the same as applying the functor then forming the complex.
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Turning an object into a cochain complex supported at zero then applying a functor is the same as applying the functor then forming the cochain complex.