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In “Arbib, Manes. Arrows, Structures and Functors. The Categorical Imperative. 6. Structured sets.” there is an approach to formalize structures. I have a strong feeling that they describe just concrete categories in other words. For a concrete category $(C, U)$, the set of structures on a set $X$ (this is Arbib, Manes's term) is $(fr\\_ob(U))^{-1}(X)$ where $fr\\_ob(U)$ is a mapping of $U$ on objects. Similarly, the set of admissible functions $A\to B$ (for $A\ B\in ob(C)$) is $im(fr\\_hom(U)(A, B))$ where $fr\\_hom(U)$ is a mapping of $U$ on morphisms.

If I am on the right track, what do other parts of Arbib, Manes's theory (e.g. optimal lifts) correspond to?

(Please add the tag “concrete-category” to this question.)

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The "ct.x" style tags are reserved for arXiv subjects. –  Qiaochu Yuan Sep 13 '11 at 16:18
    
@Qiaochu Yuan: Thanks, I did not know that. I have removed “ct.”. –  beroal Sep 16 '11 at 18:18
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