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Eduardo Ochs (Oct 17 2020 at 23:48):People, what are your favourite references on canonical subobjects? I am working on a proof for toposes of the form that becomes much simpler if I define that certain maps in are "inclusions" and I use a notation with ""s and ""s for products and coproducts of canonical subobjects... I would like to be able to say "for more on canonical subobjects see [XXX]", but I learned canonical subobjects from a brief mention to them in Lambek and Scott's book ages ago...
Morgan Rogers (he/him) (Oct 18 2020 at 10:12):A canonical subobject being a canonical representative for a particular subobjects (equivalence class of monos with the same codomain), like the inclusion of a subobject in ?
Eduardo Ochs (Oct 18 2020 at 18:37):Yes, exactly!
Fawzi Hreiki (Oct 18 2020 at 20:51):Surely the intersection and union notations cannot be used unless you’re dealing with subobjects of the same object? And if you are dealing with two subobjects of the same object, then why do they need to be canonical for you to use that notation?
Morgan Rogers (he/him) (Oct 18 2020 at 22:01):Hmm I think Johnstone also only talks about them as a passing remark along the lines that they are assumed to exist for convenience, so I wouldn't know where to look...
David Michael Roberts (Oct 18 2020 at 23:18):There was a question on MathOverflow by Kevin Buzzard whose answers might be mined for inspiration...