We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra $PV^{\omega}$ defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with $\sum_{1}^{b}$-induction over $PV^{\omega}$ and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion.