  A Hausdorff topology $\tau$ on the bicyclic monoid with adjoined zero
$\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse
semigroup topology on $\mathcal{C}^0$. We show that the lattice $\mathcal{W}$
of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the
lattice of all shift-invariant filters on $\omega$ with an attached element $1$
endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ iff
$\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate
cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved
that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and
a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a
well-ordered chain of first-countable weak topologies of order type
$\mathfrak{t}$.
