A map of simplicial modules $f : C' \rightarrow D'$ is a sequence of maps $f_i : C'_i \rightarrow D'_{d'+i}$. The new map $g : C \rightarrow D$ is the sequence of maps $g_i : C_i \rightarrow D_{d+i}$ induced by the matrix of $f_i$.
One use for this function is to get the new map of simplicial modules induced by forgetting the underlying complexes of the source and target, assuming that the source and target are obtained as Dold-Kan images.
i1 : R = ZZ/101[a,b,c];
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i2 : C = simplicialModule(freeResolution coker vars R, Degeneracy => true)
1 4 10 20
o2 = R <-- R <-- R <-- R <-- ...
0 1 2 3
o2 : SimplicialModule
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i3 : f = map(forgetComplex C, forgetComplex C, id_C)
1 1
o3 = 0 : R <--------- R : 0
| 1 |
4 4
1 : R <------------------- R : 1
{0} | 1 0 0 0 |
{1} | 0 1 0 0 |
{1} | 0 0 1 0 |
{1} | 0 0 0 1 |
10 10
2 : R <------------------------------- R : 2
{0} | 1 0 0 0 0 0 0 0 0 0 |
{1} | 0 1 0 0 0 0 0 0 0 0 |
{1} | 0 0 1 0 0 0 0 0 0 0 |
{1} | 0 0 0 1 0 0 0 0 0 0 |
{1} | 0 0 0 0 1 0 0 0 0 0 |
{1} | 0 0 0 0 0 1 0 0 0 0 |
{1} | 0 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 0 1 0 0 |
{2} | 0 0 0 0 0 0 0 0 1 0 |
{2} | 0 0 0 0 0 0 0 0 0 1 |
20 20
3 : R <--------------------------------------------------- R : 3
{0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
{1} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
o3 : SimplicialModuleMap
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i4 : assert isWellDefined f
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i5 : assert(degree f == 0)
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i6 : assert isCommutative f
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i7 : assert isSimplicialMorphism f
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i8 : normalize f --notice how the normalization is not already pruned
1 1
o8 = 0 : R <--------- R : 0
| 1 |
1 : image {0} | 0 0 0 | <----------------- image {0} | 0 0 0 | : 1
{1} | 1 0 0 | {1} | 1 0 0 | {1} | 1 0 0 |
{1} | 0 1 0 | {1} | 0 1 0 | {1} | 0 1 0 |
{1} | 0 0 1 | {1} | 0 0 1 | {1} | 0 0 1 |
2 : image {0} | 0 0 0 | <----------------- image {0} | 0 0 0 | : 2
{1} | 0 0 0 | {2} | 1 0 0 | {1} | 0 0 0 |
{1} | 0 0 0 | {2} | 0 1 0 | {1} | 0 0 0 |
{1} | 0 0 0 | {2} | 0 0 1 | {1} | 0 0 0 |
{1} | 0 0 0 | {1} | 0 0 0 |
{1} | 0 0 0 | {1} | 0 0 0 |
{1} | 0 0 0 | {1} | 0 0 0 |
{2} | 1 0 0 | {2} | 1 0 0 |
{2} | 0 1 0 | {2} | 0 1 0 |
{2} | 0 0 1 | {2} | 0 0 1 |
3 : image {0} | 0 | <------------- image {0} | 0 | : 3
{1} | 0 | {3} | 1 | {1} | 0 |
{1} | 0 | {1} | 0 |
{1} | 0 | {1} | 0 |
{1} | 0 | {1} | 0 |
{1} | 0 | {1} | 0 |
{1} | 0 | {1} | 0 |
{1} | 0 | {1} | 0 |
{1} | 0 | {1} | 0 |
{1} | 0 | {1} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{2} | 0 | {2} | 0 |
{3} | 1 | {3} | 1 |
o8 : ComplexMap
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i9 : normalize id_C
1 1
o9 = 0 : R <--------- R : 0
| 1 |
3 3
1 : R <----------------- R : 1
{1} | 1 0 0 |
{1} | 0 1 0 |
{1} | 0 0 1 |
3 3
2 : R <----------------- R : 2
{2} | 1 0 0 |
{2} | 0 1 0 |
{2} | 0 0 1 |
1 1
3 : R <------------- R : 3
{3} | 1 |
o9 : ComplexMap
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i10 : prune normalize f == normalize id_C
o10 = true
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