A characteristic property of compact support cohomology is the long exact sequence that connects the compact support cohomology groups of a space, an open subspace and its complement. Given an arbitrary invariant of algebraic varieties, taking values in a stable infinity-category C, one can wonder under what conditions it is possible to construct a “compact support” version of the invariant in such a way that this long exact sequence exists by construction. In this talk I give an answer to this question, in the form of an equivalence of categories of C-valued hypersheaves on different sites of algebraic varieties. I will discuss some applications of this theorem.

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