Sets for Mathematics question

The standard tools for analyzing an arbitrary map are the induced equivalence relation, co-equivalence relation, graph and cograph (cographs have been very frequently pictured in practice but only rarely recognized explicitly). F. William Lawvere

I would like to mention a very beautiful example of a duality, due, I believe, to Bill Lawvere. It is commonplace to note that a map f: A --> B can be identified (in any category with binary products) as a subobject G \subset A x B for which the composite G --> A x B --> A is an isomorphism. G is the graph of the morphism. Less well known (although inevitable by catgorical duality) is that a function can also be identified as a quotient Q of A + B for which the composite B --> A + B --> Q is an isomorphism. What is beautiful about this is that it reproduces our first examples of functions in which we draw some dots on the left, representing A and some on the right representing B and then some circles that include one or more elements of A and precisely one element of B. If we change "one or more" in the preceding line to "zero or more" then the set of circles constitute a partition of A + B in which each element of the partition contains precisely one element of B! Michael Barr

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