Hi - and a question on "closure operators restricted to the terminal object"
Hi! My name is Eduardo Ochs and I work more with diagrammatic reasoning than with Category Theory in the strict sense... I am currently trying to finish the last sections of this - http://angg.twu.net/math-b.html#clops-and-tops - to upload it to Arxiv. The sections that I consider finished in it should be useful to beginners that are learning topologies and closure operators.
In the sections that I am trying to finish writing down I (will) prove that in certain toposes where everything is very easy to visualize there is a bijection between closure operators on them and, aham, "closure operators restricted to the terminal object" - and this bijection is very easy to understand visually... I have all the diagrams but not all the text yet =/. I bet that this bijection is known and works on cases much more general that the ones I'm working on, but I haven't found mentions of it in the literature... any pointers would be super welcome.
UPDATE: Jens Hemelaer pointed me to this paper by Bert Lindenhovius - https://arxiv.org/abs/1405.4408v2 - and it seems that several things that I was trying to prove by myself are proved there. Time to read it and to draw its missing diagrams!