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# net(SimplicialMap) -- make a symbolic representation for a map of abstract simplicial complexes

## Synopsis

• Function: net
• Usage:
net f
• Inputs:
• Outputs:
• a net, a symbolic representation used for printing

## Description

The net of map $f \colon \Delta \to \Gamma$ between abstract simplicial complexes is a list of variables in the ring of $\Gamma$. This list determines a ring map from the ring of $\Delta$ to the ring of $\Gamma$ by sending the $i$-th variable in the ring of $\Delta$ to the $i$-th monomial on the list.

The identity map $\operatorname{id} \colon \Delta \to \Delta$ corresponds to list of variables in the ring of $\Delta$.

 i1 : S = ZZ[x_0..x_5]; i2 : Δ = simplicialComplex monomialIdeal(x_0*x_5, x_1*x_4, x_2*x_3) o2 = 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 | o2 : SimplicialComplex i3 : id_Δ o3 = | x_0 x_1 x_2 x_3 x_4 x_5 | o3 : 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 | i4 : net id_Δ o4 = | x_0 x_1 x_2 x_3 x_4 x_5 | i5 : matrix id_Δ o5 = | x_0 x_1 x_2 x_3 x_4 x_5 | 1 6 o5 : Matrix S <-- S

The next example does not come from the identity map.

 i6 : S' = ZZ[y_0..y_3]; i7 : Γ = simplicialComplex monomialIdeal(y_1*y_2) o7 = simplicialComplex | y_0y_2y_3 y_0y_1y_3 | o7 : SimplicialComplex i8 : f = map(Γ, Δ, {y_0,y_0,y_1,y_2,y_3,y_3}) o8 = | y_0 y_0 y_1 y_2 y_3 y_3 | o8 : SimplicialMap simplicialComplex | y_0y_2y_3 y_0y_1y_3 | <--- 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 | i9 : assert isWellDefined f i10 : net f o10 = | y_0 y_0 y_1 y_2 y_3 y_3 | i11 : matrix f o11 = | y_0 y_0 y_1 y_2 y_3 y_3 | 1 6 o11 : Matrix S' <-- S'