Dear Ross This discussion reminded me of your paper with Aurelio called (I think) Order Ideals. I always liked your regular categories without 1 (regular epi. monic span factorization of spans and so on). It was obvious then and now that it was a better set-up for studying "relations". I wanted to build on that approach for the Cartesian Bicategories stuff with (in those days) Max and Aurelio. Of course it was not to be. Max was far too into his own thoughts on regular categories and, in spite of being a coauthor of Order Ideals, Aurelio wanted no part of it because if you chuck 1 then you really can't call what's left a Cartesian Bicategory and the name was very important to him. But it would also be good to look at the things Bill did in his Diagonal Arguments paper to see if the absence of 1 suggests some new concepts and theorems as deep as the ones you obviously lose. Very best Richard ________________________________ From: Ross Street via Categories <categories-list@categories.org.au> Sent: June 26, 2026 03:56 To: categories-list@categories.org.au <categories-list@categories.org.au> Cc: Ross Street <ross.street@mq.edu.au> Subject: [categories] Re: Terminology for locally cartesian closed categories, with and without a terminal object CAUTION: The Sender of this email is not from within Dalhousie. 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