  Modelling and reasoning about dynamic memory allocation is one of the
well-established strands of theoretical computer science, which is particularly
well-known as a source of notorious challenges in semantics, reasoning, and
proof theory. We capitalize on recent progress on categorical semantics of full
ground store, in terms of a full ground store monad, to build a corresponding
semantics of a higher order logic over the corresponding programs. Our main
result is a construction of an (intuitionistic) BI-hyperdoctrine, which is
arguably the semantic core of higher order logic over local store. Although we
have made an extensive use of the existing generic tools, certain principled
changes had to be made to enable the desired construction: while the original
monad works over total heaps (to disable dangling pointers), our version
involves partial heaps (heaplets) to enable compositional reasoning using
separating conjunction. Another remarkable feature of our construction is that,
in contrast to the existing generic approaches, our BI-algebra does not
directly stem from an internal categorical partial commutative monoid.
