The Gluing Property

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many $\kappa$-complete ultrafilters on $\kappa$ can be glued into a $\kappa$-complete extender. We show that every $\kappa$-compact cardinal has the $2^{\kappa }$-gluing property, yet non-necessarily the $(2^{\kappa }{)}^{+}$-gluing property. Finally, we compute the exact consistency-strength for $\kappa$ to have the $\omega$-gluing property; this being $o(\kappa )={\omega }_1$.