  For a strictly semistable log scheme $Y$ over a perfect field $k$ of
characteristic $p$ we investigate the canonical \v{C}ech spectral sequence
$(C)_T$ abutting to the Hyodo-Kato (log crystalline) cohomology
$H_{crys}^*(Y/T)_{\mathbb{Q}}$ of $Y$ and beginning with the log convergent
cohomology of its various component intersections $Y^i$. We compare the
filtration on $H_{crys}^*(Y/T)_{\mathbb{Q}}$ arising from $(C)_T$ with the
monodromy operator $N$ on $H_{crys}^*(Y/T)_{\mathbb{Q}}$. We also express $N$
through residue maps and study relations with singular cohomology. If $Y$ lifts
to a proper strictly semistable (formal) scheme $X$ over a finite totally
ramified extension of $W(k)$, with generic fibre $X_K$, we obtain results on
how the simplicial structure of $X_K^{an}$ (as analytic space) is reflected in
$H_{dR}^*(X_K)=H_{dR}^*(X_K^{an})$.
