Just one more thing: remember that a locally compact space doesn't need to be compact. The same thing is happening here with local cartesian closedness. M. On 25/06/2026 23:08, Martin Escardo wrote:
The terminology "locally" has never been very lucky.
Consider e.g. "locally compact (topological) space".
(1) Some people say: every point has a compact neighbourhood.
(2) Other people say: every neighbourhood of a point has a compact neighbourhood of the point.
The approach (1) works for Hausdorff spaces only, where the condition (1) implies (2). In the absence of the Hausdorff condition, it gives spurious conclusions with no real meaning.
It is (2) that is right. But if you dig further, the right ultimate condition is that the open sets form a continuous lattice.
"locally" seems to be used as inconsistently in category theory as it is used in point-set topology.
If I could vote, I would vote for the real meaning of locality, and say that the slices satisfy the property. Any thing more or less should be added or subtracted from the terminology with further adjectives. Otherwise we get an ad hoc scaffolding of terminology.
Locality in itself, both in topology and in category theory, is an important notion. It shouldn't be distorted by the coincidences of the theorems we want to prove about it, or the examples we happen to have.
M.