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We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show that the complexities are maintained in the sense that if the functor is ${\bf{\Delta }}_\alpha ^0$ , then the interpretation that induces it is ${\rm{\Delta }}_\alpha ^{in}$ up to ${\bf{\Delta }}_\alpha ^0$ equivalence.

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Current address: DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO CANADA E-mail: maharris@math.uwaterloo.caURL:∼maharris/

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[1]Ash, C. J. and Knight, J., Computable Structures and the Hyperarithmetical Hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.
[2]Ahlbrandt, G. and Ziegler, M., Quasi-finitely axiomatizable totally categorical theories. Annals of Pure and Applied Logic, vol. 30 (1986), no. 1, pp. 6382. Stability in model theory (Trento, 1984).
[3]Baldwin, J. T., Friedman, S. D., Koerwien, M., and Laskowski, M. C., Three red herrings around Vaught’s conjecture. Transactions of the American Mathematical Society, vol. 368 (2016), no. 5, pp. 36733694.
[4]Evans, D. M. and Hewitt, P. R., Counterexamples to a conjecture on relative categoricity. Annals of Pure and Applied Logic, vol. 46 (1990), no. 2, pp. 201209.
[5]Gao, S., Invariant Descriptive Set Theory, Pure and Applied Mathematics, vol. 293, CRC Press, Boca Raton, FL, 2009.
[6]Hjorth, G., A note on counterexamples to the Vaught conjecture. Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 1, pp. 4951.
[7]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimensions in algebraic structures. Annals of Pure and Applied Logic, vol. 115 (2002), no. 1–3, pp. 71113.
[8]Hodges, W., Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.
[9]Harrison-Trainor, M., Igusa, G., and Knight, J. F., Some new computable structures of high rank. Preprint, 2016. Available at∼mattht/papers/high-scott-rank.pdf.
[10]Harrison-Trainor, M., Melnikov, A., Miller, R., and Montalbán, A., Computable functors and effective interpretability, this Journal, vol. 82 (2017), no. 1, pp. 7797.
[11]Kueker, D. W., Definability, automorphisms, and infinitary languages, The Syntax and Semantics of Infinitary Languages (Barwise, J., editor), Springer, Berlin, Heidelberg, 1968, pp. 152165.
[12]Makkai, M., An application of a method of Smullyan to logics on admissible sets. Bulletin de l’Académie polonaise des sciences. Série des sciences mathématiques, astronomiques, et physiques, vol. 17 (1969), pp. 341346.
[13]Montalbán, A., Computability theoretic classifications for classes of structures. Proceedings of ICM 2014, vol. 2 (2014), pp. 79-101.
[14]Montalbán, A., A fixed point for the jump operator on structures, this Journal, vol. 78 (2013), no. 2, pp. 425438.
[15]Montalbán, A., A robuster Scott rank. Proceedings of the American Mathematical Society, vol. 143 (2015), no. 12, pp. 54275436.
[16]Miller, R., Poonen, B., Schoutens, H., and Shlapentokh, A., A computable functor from graphs to fields. Preprint, 2015. Available at
[17]Shelah, S., Can you take Solovay’s inaccessible away? Israel Journal of Mathematics, vol. 48 (1984), no. 1, pp. 147.
[18]Solovay, R. M., A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics (2), vol. 92 (1970), pp. 156.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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