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Stream: learning: questions

Topic: Lawvere-Tierney topologies in terms of the subobject J

James Deikun (Jan 01 2024 at 15:35):

I've seen a bunch of sources (the nLab, Maclane-Moerdijk, for example) formulate the axioms for a Lawvere-Tierney topology in terms of an arrow $j : \Omega \to \Omega$ and mention that the axioms can also be specified in terms of properties of the corresponding subobject $J$, but none of them that I've looked at actually seems to include or reference such a presentation. Is there a source that has this?

Ivan Di Liberti (Jan 01 2024 at 16:05):

https://diliberti.github.io/Teaching/Teaching%20Stockholm/CatLog/2023/HL14.pdf

James Deikun (Jan 01 2024 at 17:01):

Thanks! I still have a couple of questions, though:

• On page 4 you use the representables to define the family of arrows that the inclusion of J must be orthogonal to. What happens in a general topos where you don't have the representables?
• How do the axioms $j \circ j = j$ and $j \circ \wedge = \wedge \circ (j \times j)$ arise from the orthogonality condition?