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Eric M Downes (Apr 29 2025 at 07:56):I can learn about some object using its subobject lattice, and specify much important structure this way. But there are limitations to how specific I can get.
Example: let’s say is an internal group object. We can use the topos to nicely expresses much group structure in the monoid . But the subobject lattice is a heyting algebra, is distributive. Even the normal subgroup lattice of a group is merely modular.
So to really specify an object (say, securing it within a larger diagram for overseas transport through some dizzying natural transformations) we need to either supply other unique isomorphisms and work with something like よ-embedding that preserves them, or we need to be able to work at a level of generality that permits equivalence. (Or perhaps use a forgetful functor to get to a function space where the subobjects do form a distributive lattice? At that point maybe categories aren’t the right tool though.)
What have you found to be the most high-value information that complements the subobject lattice description for specifying the unique (utui) algebraic object such-and-such. (Let’s assume to simplify matters that, within reason, we can pick our means of transport to preserve this structure.)
If it matters, by “algebraic object” I mean there’s some functor from a category with a corresponding lawvere theory containing this object in its image. If further restricting that would allow you to say something, please do so.