CatGloss

A join of the empty family (where I= I = \emptyset ) is an initial object of the category A A , as in Example X. Equivalently, it is a least element of A A : an element 0A 0 \in A such that 0a 0 \leq a for all aA a \in A . For instance, in (R,) (\reals, \mathord{\leq}) , join is supremum and there is no least element.