  Normalised multi-differential cross sections for top quark pair
($\mathrm{t\overline{t}}$) production are measured in proton-proton collisions
at a centre-of-mass energy of 13 TeV using events containing two oppositely
charged leptons. The analysed data were recorded with the CMS detector in 2016
and correspond to an integrated luminosity of 35.9 fb$^{-1}$. The
double-differential $\mathrm{t\overline{t}}$ cross section is measured as a
function of the kinematic properties of the top quark and of the
$\mathrm{t\overline{t}}$ system at parton level in the full phase space. A
triple-differential measurement is performed as a function of the invariant
mass and rapidity of the $\mathrm{t\overline{t}}$ system and the multiplicity
of additional jets at particle level. The data are compared to predictions of
Monte Carlo event generators that complement next-to-leading-order (NLO)
quantum chromodynamics (QCD) calculations with parton showers. Together with a
fixed-order NLO QCD calculation, the triple-differential measurement is used to
extract values of the strong coupling strength $\alpha_S$ and the top quark
pole mass ($m_\mathrm{T}^\text{pole}$) using several sets of parton
distribution functions (PDFs). Furthermore, a simultaneous fit of the PDFs,
$\alpha_S$, and $m_\mathrm{T}^\text{pole}$ is performed at NLO, demonstrating
that the new data have significant impact on the gluon PDF, and at the same
time allow an accurate determination of $\alpha_S$ and
$m_\mathrm{T}^\text{pole}$. The values $\alpha_S(m_\mathrm{Z})$ = 0.1135
$^{+0.0021}_{-0.0017}$ and $m_\mathrm{T}^\text{pole}$ = 170.5 $\pm$ 0.8 GeV are
extracted, which account for experimental and theoretical uncertainties, the
latter being estimated from NLO scale variations. Possible effects from Coulomb
and soft-gluon resummation near the $\mathrm{t\overline{t}}$ production
threshold are neglected in these parameter extractions.
