Biased elementary doctrines and quotient completions

In this work, we fill the gap between the elementary quotient completion introduced by Maietti and Rosolini and the exact completion of a category with weak finite limits, as described by Carboni and Vitale. To achieve this, we generalize Lawvere’s elementary doctrines to apply to categories with weak finite products, referring to these structures as biased elementary doctrines. We present two main constructions: the first, called strictification, produces an elementary doctrine from a biased one, while the second is an extension of the elementary quotient completion that generalizes the exact completion of a category with weak finite limits, even when weak finite products are involved.