The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest-dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated—namely, the spiral and funnel Rössler attractors, the Lorenz attractor, the Burke and Shaw attractor, and a four-dimensional system. A link is established with their description in terms of templates.
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