2021/11/06 by Maietti, Maria Emilia, Trotta, Davide
#03G30 #18C10 #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2111.03850
This paper aims to apply the tool of generalized existential completions of conjunctive doctrines, concerning a class Λ of morphisms of their base category, to deepen the study of regular and exact completions of existential elementary Lawvere's doctrines. After providing a characterization of generalized existential completions, we observe that both the subobjects doctrine SubC and the weak subobjects doctrine ΨC of a category C with finite limits are generalized existential completions of the constant true doctrine, the first along the class of all the monomorphisms of C while the latter along all the morphisms of C. We then name full existential completion a generalized completion of a conjunctive doctrine along the class of all the morphisms of its base. From this we immediately deduce that both the regular and the exact completion of a finite limit category are regular and exact completions of full existential doctrines since it is known that both the regular completion (D) reg / lex and the exact completion (D)ex / lex of a finite limit category D are respectively the regular completion Reg(ΨD) and the exact completion T_ΨD (as an instance of the tripos-to-topos construction) of the weak subobjects doctrine ΨD of D. Here we prove that the condition of being a generic full existential completion is also sufficient to produce a regular/exact completion equivalent to a regular/exact completion of a finite limit category. Then, we show more specialized characterizations from which we derive known results as well as remarkable examples of exact completions of full existential completions, including all realizability toposes and supercoherent localic toposes.