The alternating face map complex of a simplicial object in a preadditive category #
We construct the alternating face map complex, as a
functor alternatingFaceMapComplex : SimplicialObject C ⥤ ChainComplex C ℕ
for any preadditive category C
. For any simplicial object X
in C
,
this is the homological complex ... → X_2 → X_1 → X_0
where the differentials are alternating sums of faces.
The dual version alternatingCofaceMapComplex : CosimplicialObject C ⥤ CochainComplex C ℕ
is also constructed.
We also construct the natural transformation
inclusionOfMooreComplex : normalizedMooreComplex A ⟶ alternatingFaceMapComplex A
when A
is an abelian category.
References #
- https://stacks.math.columbia.edu/tag/0194
- https://ncatlab.org/nlab/show/Moore+complex
Construction of the alternating face map complex #
The differential on the alternating face map complex is the alternate sum of the face maps
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The chain complex relation d ≫ d
#
Construction of the alternating face map complex functor #
The alternating face map complex, on objects
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The alternating face map complex, on morphisms
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The alternating face map complex, as a functor
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The natural transformation which gives the augmentation of the alternating face map complex attached to an augmented simplicial object.
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Construction of the natural inclusion of the normalized Moore complex #
The inclusion map of the Moore complex in the alternating face map complex
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The inclusion map of the Moore complex in the alternating face map complex, as a natural transformation
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The differential on the alternating coface map complex is the alternate sum of the coface maps
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The alternating coface map complex, on objects
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The alternating face map complex, on morphisms
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The alternating coface map complex, as a functor