Isn't terminology fun? On 26 Jun 2026, at 8:08 am, Martin Escardo via Categories <categories-list@categories.org.au> wrote: The terminology "locally" has never been very lucky. I agree with Martin. The term "local" is the culprit. Here I go dragging things out of my memory. Jon Beck, Myles Tierney and others, as far back as 1968, were pushing for explicit terminology especially when the explicit term was just as short. For example, "category C is locally P" was being used to mean "each hom of C has property P". In SLNM47, Jean Bénabou introduced the local terminology for homs of bicategories. Yet Jon and Myles would prefer "small homs" to "locally small", for example. Bill Lawvere was using the term "(bi)closed bicategory" when composing with a 1-morphism had a right adjoint (so a one-object bicategory is closed iff it is as a monoidal category). In SLNM420 page 66, Brian Day says a category C with finite limits is a "closed span category" when each slice C/c is cartesian closed. I think this was a nod to both Jean and Bill by moving from the finitely complete C to the bicategory Span(C): cartesian closed slices amounts to both Span(C) closed in the Bill sense and to locally closed in the Jean sense. Having the homs of Span(C) cartesian closed when C merely has pullbacks, is also interesting. Ross