ELEMENTARY QUOTIENT COMPLETION

We extend the notion of exact completion on a category with weak nite limits to Lawvere's elementary doctrines. We show how any such doctrine admits an ele- mentary quotient completion, which is the universal solution to adding certain quotients. We note that the elementary quotient completion can be obtained as the composite of two other universal constructions: one adds eective quotients, the other forces exten- sionality of morphisms. We also prove that each construction preserves comprehension.

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