/sci/ - Science & Math

>>8405699
Sorry, my bad. I should have waited until I was fully awake, but no, I decided to post while still in bed.

It starts to make sense now. I used the pic of the isomorphims just to point to express my idea with less words. The functor G would have been a functor taking a space X to the underlying set of the Yoneda bundle, but I don't have a clue whether this would be fruitful or not.

If you want to decrease the amount of information, then how about an approach of the following sort?

Take a category [math]\textbf{C}[/math] with its objects satisfying some properties [math]P, Q[/math] and all morphisms combatible with both the properties, and a category [math]\mathbf{C}'[/math] with its objects satisfying just [math]P[/math], and for which [math]\mathbf{C}[/math] is a subcategory. If the properties are suitable, the original category is not a full subcategory, and then you can add the required morphisms to get a full subcategory with necessarily less info. Restricting the Yoneda embedding to this full subcategory would then embed a less informative version of the original category (assuming it is locally small).

I hope you reach some conclusion.