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map(SimplicialComplex,SimplicialComplex,Matrix) -- create a simplicial map between simplicial complexes

Description

A simplicial map $f: \Delta \to \Gamma$ is a function that sends the vertices of $\Delta$ to vertices of $\Gamma$, with the added condition that if $\{ v_1, v_2,..,v_k \} \in \Delta$, then $\{ f(v_1), f(v_2), ..., f(v_n) \} \in \Gamma$. If no target is specified, it is assumed that the target is the simplicial complex whose faces are $f(F)$ for all faces $F \in \Delta$. As a first example, let's look at the identity map on a 3-simplex.

 i1 : S = QQ[a,b,c,d]; i2 : Δ = simplexComplex(3,S); i3 : f = map(Δ,Δ, id_S) o3 = | a b c d | o3 : SimplicialMap simplicialComplex | abcd | <--- simplicialComplex | abcd | i4 : matrix f o4 = | a b c d | 1 4 o4 : Matrix S <-- S i5 : map f o5 = map (S, S, {a, b, c, d}) o5 : RingMap S <-- S

Here is a slightly more interesting example.

 i6 : R = QQ[s,t,u,v,w]; i7 : Γ = simplicialComplex{s*t*u,u*v*w}; i8 : g = map(Δ,Γ, {a,b,c,d,d}) o8 = | a b c d d | o8 : SimplicialMap simplicialComplex | abcd | <--- simplicialComplex | uvw stu | i9 : source g o9 = simplicialComplex | uvw stu | o9 : SimplicialComplex i10 : target g o10 = simplicialComplex | abcd | o10 : SimplicialComplex i11 : image g o11 = simplicialComplex | cd abc | o11 : SimplicialComplex