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BOREL FUNCTORS AND INFINITARY INTERPRETATIONS

Published online by Cambridge University Press:  05 October 2018

MATTHEW HARRISON-TRAINOR ,
RUSSELL MILLER  and
ANTONIO MONTALBÁN
MATTHEW HARRISON-TRAINOR
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA BERKELEY, CA, USA
RUSSELL MILLER
Affiliation:
MATHEMATICS DEPT., QUEENS COLLEGE PH.D. PROGRAMS IN MATHEMATICS & COMPUTER SCIENCE GRADUATE CENTER, CITY UNIVERSITY OF NEW YORK NEW YORK, NY, USAE-mail:russell.miller@qc.cuny.eduURL: http://qcpages.qc.cuny.edu/∼rmiller
ANTONIO MONTALBÁN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA, USAE-mail:antonio@math.berkeley.eduURL: www.math.berkeley.edu/∼antonio
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Abstract

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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.

Keywords

03D4503C75interpretationsBorel functorsautomorphism groups
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1434 - 1456
Copyright
Copyright © The Association for Symbolic Logic 2018 

Footnotes

*

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

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