The paper is now wrapped. And is also on the ArXiv. I think the main part is good especially the beginning. Later, I did not put much time in condensing things. I’m very excited how convenient it is to work with the objects. It seems that we get a nice topos of semi-simplicial sets in which each element is nicely equipped with a topology. In the video, I outline the main definitions and look at a few examples.
By the way, here is the code listed in that paper.
F[G_]:=Module[{l=Map[Length,G]},If[G=={},{},
Table[Sum[If[l[[j]]==k,1,0],{j,Length[l]}],{k,Max[l]}]]]; s[x_]:=Signature[x];L=Length;
s[x_,y_]:=If[SubsetQ[x,y]&&(L[x]==L[y]+1),s[Prepend[y,Complement[x,y][[1]]]]*s[x],0];
Dirac[G_]:=Module[{f=F[G],b,d,n=Length[G]},b=Prepend[Table[Sum[f[[l]],{l,k}],{k,Length[f]}],0];
d=Table[s[G[[i]],G[[j]]],{i,n},{j,n}]; {d+Transpose[d],b}];
Hodge[G_]:=Module[{Q,b,H},{Q,b}=Dirac[G];H=Q.Q;Table[Table[H[[b[[k]]+i,b[[k]]+j]],
{i,b[[k+1]]-b[[k]]},{j,b[[k+1]]-b[[k]]}],{k,Length[b]-1}]]; Betti[G_]:=Map[nu,Hodge[G]];
Closure[A_]:=If[A=={},{},Delete[Union[Sort[Flatten[Map[Subsets,A],1]]],1]];
Whitney[s_]:=If[Length[EdgeList[s]]==0,Map[{#}&,VertexList[s]], Map[Sort,Sort[Closure[
FindClique[s,Infinity,All]]]]]; nu[A_]:=If[A=={},0,Length[NullSpace[A]]];
Shannon[A_,B_]:=Module[{q=Max[Flatten[A]],Q,G={}}, Q=Table[B[[k]]+q,{k,Length[B]}];
Do[G=Append[G,Sort[Union[A[[a]],Q[[b]]]]],{a,Length[A]},{b,Length[Q]}]; Sort[G]];
OpenStar[G_,x_]:=Module[{U={}},Do[If[SubsetQ[G[[k]],x],U=Append[U,G[[k]]]],{k,Length[G]}];U];
Basis[G_]:=Table[OpenStar[G,G[[k]]],{k,Length[G]}]; Stars=Basis;
RandomOpenSet[G_,k_]:=Module[{A=RandomChoice[Basis[G],k],U={}},Do[U=Union[U,A[[j]]],{j,k}];U];
Betti[G_,U_,K_]:={"b_G=",Betti[G],"b_U=",Betti[U],"b_K=",Betti[K]};
G=Whitney[RandomGraph[{20,50}]]; U=RandomOpenSet[G,10];K=Complement[G,U]; Print[Betti[G,U,K]];
S={{1},{2,3}}; G=Shannon[Shannon[S,S],Shannon[S,S]]; Print["Betti(T^4) unshifted",Betti[G] ]
S2={{1},{2,3,4}}; G=Shannon[S2,S2]; Print["Betti S^2 x S^2 (not yet shifted)", Betti[G]]
