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Is there a notion of harmonic forms and Hodge theory for Sullivan's piecewise smooth differential forms on a simplicial set? Let me recall some background. Hodge Theory on a Riemannian manifold A ...

For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices $$ \Phi : A \to B \otimes \Omega^*(\Delta^n) $$ and simplices maps ...