The Conley index is a spatial refinement of the Morse index. Informally speaking, it is a ‘space’ that describes the local dynamics around an isolated invariant subset of a topological dynamical system. In this talk, I will explain a new formulation of Conley index theory, which I think is simpler and more flexible than the traditional formulation. One important point is that the Conley index should be defined as a based equivariant condensed set/anima, not as a mere homotopy type of topological spaces. Beside that, our formulation features two classes of maps – open embeddings and proper maps – of locally compact Hausdorff spaces, which makes us tempted to speculate that Conley index theory might be related to the six-functor formalism in some way.

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