Description
The identity map on the underlying vertex set of an abstract simplicial complex induces the identity map on the entire complex.
The first example is the identity map on a $4$-simplex.
i1 : S = ZZ[a..e];
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i2 : Δ = simplexComplex(4, S)
o2 = simplicialComplex | abcde |
o2 : SimplicialComplex
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i3 : f = id_Δ
o3 = | a b c d e |
o3 : SimplicialMap simplicialComplex | abcde | <--- simplicialComplex | abcde |
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i4 : assert (isWellDefined f and source f === Δ and
target f === Δ and matrix f === vars S)
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The second example is the identity map on the octahedron.
i5 : R = ZZ[x_0..x_5];
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i6 : Γ = simplicialComplex monomialIdeal(x_0*x_5, x_1*x_4, x_2*x_3)
o6 = simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 |
o6 : SimplicialComplex
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i7 : g = id_Γ
o7 = | x_0 x_1 x_2 x_3 x_4 x_5 |
o7 : SimplicialMap simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | <--- simplicialComplex | x_3x_4x_5 x_2x_4x_5 x_1x_3x_5 x_1x_2x_5 x_0x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 |
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i8 : assert (isWellDefined g and source g === Γ and
target g === Γ and matrix g === vars R)
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