# Every mathematical object can be determinated uniquelly by his categorical structure? [closed]

I try to understand the Grothendieck philosophy. In the Mochizuki´s paper Categorical Representation of Local Noetherian Log Schemes, Mochizuki characterizes an algebraic variety using the subcategory of the finite type morphisms in the comma category. I want know if it result can be generalizated for a arbitrary category, using the comma category and as Theo Johnson-Freyd notice the forgetfull funtor.

• Your intuition cannot be quite correct. What happens when the ambient category consists of two points with no nontrivial morphisms? I think a correct version of Yoneda says that an object $X$ in a category $\mathcal C$ can be recovered from the comma category $\mathcal C_{/X}$ together with the forgetful functor $\mathcal C_{/X} \to \mathcal C$. – Theo Johnson-Freyd Jun 5 '17 at 21:05
• The published version of the paper is freely available at doi.org/10.1016/j.aim.2003.10.005 – Philippe Gaucher Jun 6 '17 at 8:05