Text to accompany “ Cell Groups Reveal Structure of Stimulus Space ”

simplicial complex. An abstract simplicial complex is a set {1, ..., N} with a set K of subsets such that if Y ∈ K and X ⊂ Y then X ∈ K. In other words, every subset of a subset in K is also included in K. Every simplicial complex is an abstract simplicial complex, where the elements of the set {1, ..., N} correspond to labelled vertices, and the set of subsets K corresponds to simplices (together with their faces, which are also simplices). Conditions (i) and (ii) for a simplicial complex guarantee that every subset of a subset in K is also in K. 2 Simplicial homology with real coefficients There are many definitions of homology groups, and these can be shown to be equivalent for many spaces of interest [Hatcher, 2002]. We need only compute homology for two kinds of topological spaces: simplicial complexes (obtained from cell groups) and our two-dimensional environments. Simplicial homology with real coefficients is comparatively easy to define, and suffices for our purposes. Chains. Let K = {σ i } be a simplicial complex with simplices σ i , where k denotes the simplex dimension. A simplicial k-chain is a formal sum of k-dimensional simplices Nk ∑ i=1 ciσ k i , where ci ∈ R and Nk is the number of k-dimensional simplices in K. Let Ck denote the group of all possible k-chains on K. Since our coefficients are in R, Ck is just the vector space Rk .