2017/10/06 by Chen, Ruiyuan
#03C15 #03E15 #03G30 #18C10 #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1710.02246
We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic \mathcal Lω1ω: every countable \mathcal Lω1ω-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (\mathcal L, \mathcal T) and (\mathcal L', \mathcal T') (in possibly different languages \mathcal L, \mathcal L'), every Borel functor Mod(\mathcal L', \mathcal T') → Mod(\mathcal L, \mathcal T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some \mathcal L'ω1ω-interpretation of \mathcal T in \mathcal T'. This generalizes a recent result of Harrison-Trainor, Miller, and Montalbán in the case where \mathcal T, \mathcal T' each have a single countable model up to isomorphism.