  We undertake the first comprehensive and quantitative real-space analysis of
the cosmological information content in the environments of the cosmic web
(voids, filaments, walls, and nodes) up to non-linear scales, $k = 0.5$
$h$/Mpc. Relying on the large set of $N$-body simulations from the Quijote
suite, the environments are defined through the eigenvalues of the tidal tensor
and the Fisher formalism is used to assess the constraining power of the power
spectra derived in each of the four environments and their combination. Our
results show that there is more information available in the
environment-dependent power spectra, both individually and when combined all
together, than in the matter power spectrum. By breaking some key degeneracies
between parameters of the cosmological model such as
$M_\nu$--$\sigma_\mathrm{8}$ or $\Omega_\mathrm{m}$--$\sigma_8$, the power
spectra computed in identified environments improve the constraints on
cosmological parameters by factors $\sim 15$ for the summed neutrino mass
$M_\nu$ and $\sim 8$ for the matter density $\Omega_\mathrm{m}$ over those
derived from the matter power spectrum. We show that these tighter constraints
are obtained for a wide range of the maximum scale, from $k_\mathrm{max} = 0.1$
$h$/Mpc to highly non-linear regimes with $k_\mathrm{max} = 0.5$ $h$/Mpc. We
also report an eight times higher value of the signal-to-noise ratio for the
combination of spectra compared to the matter one. Importantly, we show that
all the presented results are robust to variations of the parameters defining
the environments hence suggesting a robustness to the definition we chose to
define them.
