..."An example of a concrete model of Intuitionistic Linear Logic is the category
of cpos and strict continuous functions. A cpo is a partial order with the
property that each directed subset has a least upper bound. Note that this
entails that a cpo has a bottom element. A monotone function between cpos
is continuous if it preserves the least upper bound of any non-empty directed
subset, and it is strict if it preserves the bottom element. The symmetric
monoidal structure is given by the smash product, the internal-hom of two
objects is given by the set of strict continuous functions with the pointwise
order, and the comonad is given by the lift operation."

?? .... initial object as directed colimit ??? ..... ???? .... ?? convention ?? .... ??? ..... ?????? ..... ???? ....... ??? ......... ?? maybe "directed" vs "filtered" ?? ....

?? well, i think i see the conceptual motivation for why empty category shouldn't count as directed/filtered .... ??? no object with cocone from empty diagram ?? ...

?? not sure yet whether i see conceptual motivation for opposite convention ... ??? ...