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Topological endomorphism monoids of models of geometric theories

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Morgan Rogers

A classical model theory result states that for a (set-theoretic) model of a first-order theory, there is a Galois connection between subgroups of the automorphism group of the model and `relational extensions' of the model, and the subgroups which are fixed by this connection are precisely the closed subgroups for the `pointwise convergence topology' on the automorphism group. We prove an analogous result for endomorphism monoids of models, grounded in the theory of classifying toposes. In particular, we show that the topos of continuous actions of the endomorphism monoid with respect to the pointwise convergence topology classifies a natural theory associated to the model.

Keywords:
model theory, classifying topos, Grothendieck topos

2020 MSC:
03C55, 18B25, 18B40, 18C10, 18F10

*Theory and Applications of Categories,*
Vol. 42, 2024,
No. 3, pp 41-58.

Published 2024-06-21.

http://www.tac.mta.ca/tac/volumes/42/3/42-03.pdf

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