September 18
Hans Schoutens,
CUNY
The benign logic of equivalent structures
In my previous talk (Spring 26), I gave a tentative description of non-evil sentences in categories. When sorting out the details, I realized that this can be set within a much more abstract, model-theoretic framework. Given a first-order language $L$, a subset $\Lambda$ of $L$-formulas is a logic, if it is closed under conjunction, negation, quantification and term substitution. It turns out that many of the standard constructions carry over to these fragments: compactness of Stone spaces (although they can now be finite!), completions, and in particular, the analogue of a weak form of the Keisler-Shelah theorem: two structures satisfy the same $\Lambda$-sentences if an only if there is a map from one to an ultrapower of the other that preserves all $\Lambda$-formulas. Assume in addition that $L$ contains a binary predicate $E$ and only consider structures in which $E$ yields a congruence (equivalence) relation. Call two such structures equivalent if there are homomorphisms in either direction whose compositions are congruent (point-wise) to the identity map. Using this, we can now define the logic consisting of the so-called benign formulas. In particular, a sentence is benign if and only if equivalent structures agree on it. To derive some meaningful properties, we also need to assume that the quotient map, i.e., the map sending an element to its congruence class, is split. I'll give some examples, including the case of categories, where this machinery can be applied.