On multiplicative spectral sequences for nerves and the free loop spaces

We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a topological category. By applying the machinery to a Borel construction, we determine explicitly the mod $p$ cohomology algebra of the free loop space of the real projective space for each odd prime $p$. This is highlighted as an important computational example of such a spectral sequence. Moreover, we try to represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold $M$ with differential forms on the universal cover of $M$ via Chen's iterated integral map.