A stratified approach to Betti cohomology

Let $X$ be a complex algebraic variety. For each non-negative integer $n$, one may consider the (cohomology of the sheaf of) functions defined on a formal neighbourhood of the small diagonal in the $n$-fold self product of $X$. As $n$ varies, these objects fit together to give a chain complex. More intrinsically, this complex computes the cohomology of the structure sheaf on the stratifying site defined by $X$. In my talk, I will explain why the above complex computes the Betti cohomology of $X$. This identification was conjectured by Grothendieck, and proven by him for smooth varieties. Our approach passes through derived de Rham cohomology.