Intuitively, we know that partitions of a set X and equivalence relations on X contain the same information. What does it mean, in general, for two mathematical structures to contain the same information? One can say that the set of partitions on X is bijective to the set of equivalence relations on X. However, there is a certain sense in which this bijection is ‘natural’. Category theory explores this idea through a notion of ‘natural transformation’ introduced already in the first paper on the subject. In this Thesis we propose a different approach, which is based on set-theoretic representations of mathematical structures. Comparison of the two approaches is left for a future work.
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