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1029 Define $\cl{\cC}(W)$ as a vector space via |
1029 Define $\cl{\cC}(W)$ as a vector space via |
1030 \[ |
1030 \[ |
1031 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1031 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1032 \] |
1032 \] |
1033 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1033 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1034 Elements of a summand indexed by an $m$-sequences will be call $m$-simplices. |
1034 Elements of a summand indexed by an $m$-sequence will be call $m$-simplices. |
1035 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1035 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1036 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1036 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1037 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1037 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1038 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1038 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1039 \[ |
1039 \[ |