I think the following are also examples: - the category of Sets and monomorphisms. More generally, if C is an lccc (in the weaker sense), and (E,M) is a factorisation system on C for which E-maps are stable under pullbacks along M-maps, then I think the category of C-objects and M-maps is an lccc (in the weaker sense), but will normally not have a terminal object. - the category of arrows in Set and pullback squares. More generally, if C is an lccc (in the weaker sense) and E ---> C is a fibration, then I think E_cart (E-objects and cartesian maps) is lccc (in the weaker sense) but will normally not have a terminal object. Richard George Janelidze via Categories <categories-list@categories.org.au> writes:
Dear Peter,
There are many examples of Set-valued Galois theories where covering morphisms compose and locally form presheaf toposes. Or, note that the category of 'small' categories with morphisms being discrete fibrations is in the same relation with the presheaf toposes as the category of topological spaces and local homeomorphisms with the toposes of sheaves over topological spaces. Therefore, I would not say "only one not-wholly-artificial example".
But I agree with your suggestion, and I would add the question: do we want every groupoid to be locally cartesian closed?
Best regards, George
------------------------------------------------------------------------------- From: P.T. Johnstone via Categories <categories-list@categories.org.au> Sent: Thursday, June 25, 2026 10:13 To: Michael Shulman via Categories <categories-list@categories.org.au> Cc: Emily Riehl <eriehl@jhu.edu>; Jon Sterling <jon@jonmsterling.com>; Michael Shulman <shulman@sandiego.edu>; P.T. Johnstone <ptj1000@cam.ac.uk> Subject: [categories] Re: Terminology for locally cartesian closed categories, with and without a terminal object
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Sorry to come late to this — I've been away for a couple of days.
When I wrote the first two parts of the Elephant, I thought quite a lot about terminological questions like this. For "locally cartesian closed", I decided to break the usual rule that "locally P" means "all slices have property P" but not that the category itself has P. The reason was that there is, as far as I know, only one not-wholly-artificial example of a category whose slices are all cartesian closed but which is not itself cc (namely the category of all topological spaces and local homeomorphisms); and it didn't seem worth burdening oneself with more complicated terminology for the sake of this one counterexample.
Peter Johnstone ------------------------------------------------------------------------------- From: Michael Shulman via Categories <categories-list@categories.org.au> Sent: 24 June 2026 10:53 PM To: Jon Sterling via Categories <categories-list@categories.org.au> Cc: Emily Riehl <eriehl@jhu.edu>; Jon Sterling <jon@jonmsterling.com>; Michael Shulman <shulman@sandiego.edu> Subject: [categories] Re: Terminology for locally cartesian closed categories, with and without a terminal object
Somewhat wordy, but one could say "cartesian and locally cartesian closed category" or "globally and locally cartesian closed category" for one with a terminal object.
On Wed, Jun 24, 2026 at 2:07 PM Jon Sterling via Categories <categories-list@categories.org.au> wrote:
Hi Emily,
Just a little thought. Personally I prefer that the definition leave out a terminal object, although I am sure that it may be possible to find examples of me in print having assumed it without mentioning it. My reason is twofold: first, consistency with one of the two main uses of "locally" in category theory; second, I think that even in type theory we over-emphasise the importance of the empty context (which does not arise in the use of dependent type theory, where all language is understood as taking place in some undetermined context anyway for the sake of substitution stability — and dependently typed internal languages can be used in semantic settings even when you don't have a terminal object).
As for a good name for lccc w/ terminal object, that I don't have an opinion on :)
Best, Jon
On Jun 24, 2026, at 2:19 PM, Emily Riehl via Categories <categories-list@categories.org.au> wrote:
There has been some discussion on the Category Theory zulip about the proper terminology for locally cartesian closed categories that I’d like to extend to the broader community.
I raised this question because I have added a new section on locally cartesian closed categories as part of a revised second edition of Category Theory in Context. The current draft, can be found here and comments on this or anything else are very welcome (especially before the end of August, with sooner better than later):
https://emilyriehl.github.io/files/context.pdf⚠️
Does the community prefer reserving the term “locally cartesian closed categories” for the case where a terminal object exists (in addition to pullbacks)? If so, is there a good name for the version that is missing the terminal object? If not, is there a good name for the version that has a final limits?
As pointed out on the chat (h/t Nathanael Arkor) both conventions are widely used in the literature.
Thanks, Emily
PS: I was personally persuaded of the utility of the more general notion by the example of sets and monomorphisms. In the current draft of the text, I’m trying to provide intuition for some of the connections to logic without explicitly mentioning dependent type theory.
-- Kelly Miller Professor of Mathematics (she/her) Johns Hopkins University emilyriehl.github.io
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