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Reid Barton (Jun 01 2021 at 15:05):Is there a "standard" term for a category which is both adhesive and extensive (or equivalently, adhesive with a strict initial object)?
Here by "standard" I mean with at least one pre-existing usage.
Reid Barton (Jun 01 2021 at 15:08):(Rationale: Any such category has a cd-structure whose distinguished squares are the pushouts of monomorphisms; what I really want is a name for the topology it generates. But this topology includes the empty family as a covering family of the initial object, and so it doesn't seem quite appropriate to call it the "adhesive topology", since an adhesive category doesn't have to have an initial object.)
Fawzi Hreiki (Jun 01 2021 at 17:07):Sticky?
Reid Barton (Jun 01 2021 at 18:21):LOL, I thought about "adherent" (to imitate "coherent") but indeed there are many other synonyms for "adhesive" to consider!
Fawzi Hreiki (Jun 01 2021 at 21:04):Do you have any examples of adhesive categories which aren't extensive. I see on the nLab page that any adhesive category with a strict initial object is extensive so I imagine most adjesive categories are extensive.
Reid Barton (Jun 01 2021 at 22:40):Apparently pointed sets is an example
Spencer Breiner (Jun 01 2021 at 23:24):When you pull back along the coproduct inclusions, you pick up two copies of the fiber over the distinguished point, but when you add them back together, only the distinguished points get re-merged. This is an interesting example since the pullbacks are coproduct inclusions, but not for the same coproduct decompositions.
Fawzi Hreiki (Jun 02 2021 at 02:53):Oh nice. That’s a good one.