Lecture 2: Clarification of generalized elements


Brendan mentioned shapes could be used to probe other objects. For the Set category there is a special terminal object (the 1-point, a singleton set) that can be used to point all the elements of a set. One might use n-points instead of singletons but it turns out sets are determined completely by 1-points (since n-points can be recursively obtained from 1-points) . Also, we know this is not true for all categories: knowing about the points of an object is insufficient to count as knowing its elements (construed broadly).

I'd like to know if is possible (and how) to determine the minimal shape collection required to know a category. I haven't watched beyond lecture 2, so there might be something I am missing.

Also, would you mind giving an example of how the dual of probe ("observer") operates? Shapes in the context of generalized elements operate as imaging instruments for categories, but I am not sure about the interpretation of the dual case where what you change is the object under study and not the shape.


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