The Dold-Kan Correspondence in Macaulay2 -- compute the image of a non-negatively graded complex under the Dold-Kan correspondence

Usage:

simplicialModule(C,d)

simplicialModule(C)
Inputs:
- C, a complex,
- d, an integer, an integer specifying the top degree of the output SimplicialModule (if d is not specified, the top degree is the length of the complex C)

Given any non-negatively graded chain complex, the Dold-Kan correspondence is an equivalence of categories: $$\text{Ch}_{\geq 0} (R) \leftrightarrow \text{Simplicial R-modules}.$$ If the simplicial module is obtained as a Dold-Kan image of some complex $C$, then for each $i$ there is a decomposition $$S_i = C_0 \oplus C_1^{\binom{i}{1}} \oplus \cdots \oplus C_j^{\binom{i}{j}} \oplus \cdots.$$ To be technically correct, each of the direct summands $C_j$ of $S_i$ should be thought of as being parametrized by an order preserving surjection $f : [i] \to [j]$ (where $[n] := \{ 0 ,\dots , n \}$). To deduce which surjection corresponds to a given summand, first notice that order preserving surjections $f : [i] \to [j]$ are in bijection with compositions of $i+1$ into $j+1$ parts by just listing the sizes of the fibers of the map $f$. For instance, the composition $(2,2,1)$ corresponds to the surjection $f : [4] \to [2]$ defined via $$f(0) = f(1) = 0, \quad f(2) = f(3) = 1, \quad f(4) = 2.$$ With this in mind, one can deduce the surjection corresponding to a summand as follows:

i1 : R = ZZ/101[a..c]

o1 = R

o1 : PolynomialRing

i2 : C = koszulComplex vars R

      1      3      3      1
o2 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o2 : Complex

i3 : S = simplicialModule(C, 4, Degeneracy => true)

      1      4      10      20      35
o3 = R  <-- R  <-- R   <-- R   <-- R  <-- ...
                                    
     0      1      2       3       4

o3 : SimplicialModule

i4 : S_2

      10
o4 = R

o4 : R-module, free, degrees {0..1, 5:1, 3:2}

i5 : components S_2 --these are all the modules showing up

       1   3   3   3
o5 = {R , R , R , R }

o5 : List

i6 : S_(2,1)

      6
o6 = R

o6 : R-module, free, degrees {6:1}

i7 : components S_(2,1) --these are only the components of C_1^2

       3   3
o7 = {R , R }

o7 : List

i8 : sort(select(compositions(2,3), i -> all(i, j -> j>0)))

o8 = {{1, 2}, {2, 1}}

o8 : List

The ordering of the surjections corresponding to a summand agrees with the lexicographic ordering of compositions via the bijection mentioned above. Thus the first summand of $C_1$ appearing in $S_2$ corresponds to the surjection $f : [2] \to [1]$ given by $f(0) = 0$ and $f(1) = f(2) = 1$, and the second summand corresponds to the surjection $f: [2] \to [1]$ with $f(0) = f(1) = 0$ and $f(2) = 1$. These surjections may be accessed more directly using the summandSurjection command:

i9 : summandSurjection(2,0) --the surjection parametrizing C_0

o9 = {HashTable{0 => 0}}
                1 => 0
                2 => 0

o9 : List

i10 : summandSurjection(2,1) --the surjections parametrizing C_1

o10 = {HashTable{0 => 0}, HashTable{0 => 0}}
                 1 => 1             1 => 0
                 2 => 1             2 => 1

o10 : List

i11 : summandSurjection(2,2) --the surjection parametrizing C_2

o11 = {HashTable{0 => 0}}
                 1 => 1
                 2 => 2

o11 : List

Notice that the individual terms $C_j^{\binom{i}{j}}$ may be accessed as the $(i,j)$-component of the simplicial module. There are moreover explicit formulas for the face and degeneracy maps in terms of the differentials of $C$. These can also be accessed:

i12 : S.dd --face maps

                1                   4
o12 = (0, 0) : R  <--------------- R  : (1, 0)
                     | 1 a b c |

                1                   4
      (0, 1) : R  <--------------- R  : (1, 1)
                     | 1 0 0 0 |

                4                                      10
      (1, 0) : R  <---------------------------------- R   : (2, 0)
                     {0} | 1 a b c 0 0 0 0  0  0  |
                     {1} | 0 0 0 0 1 0 0 -b -c 0  |
                     {1} | 0 0 0 0 0 1 0 a  0  -c |
                     {1} | 0 0 0 0 0 0 1 0  a  b  |

                4                                   10
      (1, 1) : R  <------------------------------- R   : (2, 1)
                     {0} | 1 0 0 0 0 0 0 0 0 0 |
                     {1} | 0 1 0 0 1 0 0 0 0 0 |
                     {1} | 0 0 1 0 0 1 0 0 0 0 |
                     {1} | 0 0 0 1 0 0 1 0 0 0 |

                4                                   10
      (1, 2) : R  <------------------------------- R   : (2, 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 |

                10                                                              20
      (2, 0) : R   <---------------------------------------------------------- R   : (3, 0)
                      {0} | 1 a b c 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 -b -c 0  0  0  0  0 0 0 0  |
                      {1} | 0 0 0 0 0 1 0 0 0 0 a  0  -c 0  0  0  0 0 0 0  |
                      {1} | 0 0 0 0 0 0 1 0 0 0 0  a  b  0  0  0  0 0 0 0  |
                      {1} | 0 0 0 0 0 0 0 1 0 0 0  0  0  -b -c 0  0 0 0 0  |
                      {1} | 0 0 0 0 0 0 0 0 1 0 0  0  0  a  0  -c 0 0 0 0  |
                      {1} | 0 0 0 0 0 0 0 0 0 1 0  0  0  0  a  b  0 0 0 0  |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  1 0 0 c  |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 1 0 -b |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0 0 1 a  |

                10                                                       20
      (2, 1) : R   <--------------------------------------------------- R   : (3, 1)
                      {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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 1 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 0 0 0 1 0 0 1 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 |

                10                                                       20
      (2, 2) : R   <--------------------------------------------------- R   : (3, 2)
                      {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 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 |

                10                                                       20
      (2, 3) : R   <--------------------------------------------------- R   : (3, 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 |
                      {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 |

                20                                                                                                 35
      (3, 0) : R   <--------------------------------------------------------------------------------------------- R   : (4, 0)
                      {0} | 1 a b c 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  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 -b -c 0  0  0  0  0  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 a  0  -c 0  0  0  0  0  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  a  b  0  0  0  0  0  0  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  -b -c 0  0  0  0  0 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  a  0  -c 0  0  0  0 0 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  a  b  0  0  0  0 0 0 0 0 0 0 0 0 0  0  0  0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0  0  0  0  0  0  -b -c 0  0 0 0 0 0 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  a  0  -c 0 0 0 0 0 0 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  a  b  0 0 0 0 0 0 0 0 0 0  0  0  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  1 0 0 0 0 0 0 0 0 c  0  0  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 1 0 0 0 0 0 0 0 -b 0  0  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 1 0 0 0 0 0 0 a  0  0  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 1 0 0 0 0 0 0  c  0  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0 1 0 0 0 0 0  -b 0  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0 0 1 0 0 0 0  a  0  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 1 0 0 0  0  c  0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 1 0 0  0  -b 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 1 0  0  a  0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0  0  0  1 |

                20                                                                                     35
      (3, 1) : R   <--------------------------------------------------------------------------------- R   : (4, 1)
                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 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 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 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 |
                      {2} | 0 0 0 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 |
                      {2} | 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 |

                20                                                                                     35
      (3, 2) : R   <--------------------------------------------------------------------------------- R   : (4, 2)
                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 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 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 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 0 0 0 0 0 0 0 0 0 |
                      {1} | 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 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 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 0 |
                      {2} | 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 |
                      {2} | 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 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 0 0 0 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 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 |

                20                                                                                     35
      (3, 3) : R   <--------------------------------------------------------------------------------- R   : (4, 3)
                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 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 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 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 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 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 0 0 0 0 0 0 0 |
                      {1} | 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 0 0 0 0 0 0 |
                      {1} | 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 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 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 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 |
                      {2} | 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 |
                      {2} | 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 |
                      {2} | 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 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 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 |

                20                                                                                     35
      (3, 4) : R   <--------------------------------------------------------------------------------- R   : (4, 4)
                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 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 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 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 0 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 0 0 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 0 0 0 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 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |

o12 : SimplicialModuleMap

i13 : S.ss --degeneracy maps

                4                 1
o13 = (1, 0) : R  <------------- R  : (0, 0)
                     {0} | 1 |
                     {1} | 0 |
                     {1} | 0 |
                     {1} | 0 |

                10                       4
      (2, 0) : R   <------------------- R  : (1, 0)
                      {0} | 1 0 0 0 |
                      {1} | 0 0 0 0 |
                      {1} | 0 0 0 0 |
                      {1} | 0 0 0 0 |
                      {1} | 0 1 0 0 |
                      {1} | 0 0 1 0 |
                      {1} | 0 0 0 1 |
                      {2} | 0 0 0 0 |
                      {2} | 0 0 0 0 |
                      {2} | 0 0 0 0 |

                10                       4
      (2, 1) : R   <------------------- R  : (1, 1)
                      {0} | 1 0 0 0 |
                      {1} | 0 1 0 0 |
                      {1} | 0 0 1 0 |
                      {1} | 0 0 0 1 |
                      {1} | 0 0 0 0 |
                      {1} | 0 0 0 0 |
                      {1} | 0 0 0 0 |
                      {2} | 0 0 0 0 |
                      {2} | 0 0 0 0 |
                      {2} | 0 0 0 0 |

                20                                   10
      (3, 0) : R   <------------------------------- R   : (2, 0)
                      {0} | 1 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 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 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 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 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 |

                20                                   10
      (3, 1) : R   <------------------------------- R   : (2, 1)
                      {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 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 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 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 |

                20                                   10
      (3, 2) : R   <------------------------------- R   : (2, 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 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 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 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 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 |

                35                                                       20
      (4, 0) : R   <--------------------------------------------------- R   : (3, 0)
                      {0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 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 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 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 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

                35                                                       20
      (4, 1) : R   <--------------------------------------------------- R   : (3, 1)
                      {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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

                35                                                       20
      (4, 2) : R   <--------------------------------------------------- R   : (3, 2)
                      {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 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

                35                                                       20
      (4, 3) : R   <--------------------------------------------------- R   : (3, 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 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {1} | 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

o13 : SimplicialModuleMap

i14 : S.dd_(2,0)

o14 = {0} | 1 a b c 0 0 0 0  0  0  |
      {1} | 0 0 0 0 1 0 0 -b -c 0  |
      {1} | 0 0 0 0 0 1 0 a  0  -c |
      {1} | 0 0 0 0 0 0 1 0  a  b  |

              4      10
o14 : Matrix R  <-- R

If you want to restrict/project a face/degeneracy map to a particular summand group, this can be done using the array notation. The $k$-th summand group of $S_i$ corresponds to $C_k^{\binom{i}{k}}$:

i15 : (S.dd_(2,0))_[1]^[0]  --restrict to C_1 summand group of S_2, project onto C_0 summand of S_1

o15 = | a b c 0 0 0 |

              1      6
o15 : Matrix R  <-- R

i16 : (S.dd_(2,0))_[1]^[1]  --restrict to C_1 summand group of S_2, project onto C_1 summand of S_1

o16 = {1} | 0 0 0 1 0 0 |
      {1} | 0 0 0 0 1 0 |
      {1} | 0 0 0 0 0 1 |

              3      6
o16 : Matrix R  <-- R

i17 : (S.dd_(2,0))_[2]^[1]  --restrict to C_2 summand of S_2, project onto C_1 summand of S_1

o17 = {1} | -b -c 0  |
      {1} | a  0  -c |
      {1} | 0  a  b  |

              3      3
o17 : Matrix R  <-- R

The first computation tells us that the component of the face map $d_{2,0}$ mapping $$C_1^{\oplus 2} \to C_0$$ is given by the differential of the original complex $C$ applied to the first copy of $C_1$ (we use the compositions corresponding to the surjections to label the free modules). The second computation shows the component $C_1^{\oplus 2} \to C_1$ is simply the identity map on the second copy of $C_1$. The third computation shows the component $$C_2 \to C_1$$ is also given by the differential of $C$.

The Dold-Kan Correspondence in Macaulay2 -- compute the image of a non-negatively graded complex under the Dold-Kan correspondence

See also