The gluing property

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal [Formula: see text] and a cardinal [Formula: see text], we say that [Formula: see text] has the [Formula: see text]-gluing property if every sequence of [Formula: see text]-many [Formula: see text]-complete ultrafilters on [Formula: see text] can be glued into an extender. We show that every [Formula: see text]-compact cardinal has the [Formula: see text]-gluing property, yet non-necessarily the [Formula: see text]-gluing property. Finally, we compute the exact consistency strength for [Formula: see text] to have the [Formula: see text]-gluing property — this being [Formula: see text].