Mathematics Colloquium

Thursday, October 19, 2017
4:20 PM - 5:20 PM (ET)
Exley Science Center Tower ESC 121
Event Type
Canalia, Caryn

Speaker: Alex Kruckman, Indiana University-Bloomington

Title: First-order logic and cologic over a category

Abstract: In ordinary first-order logic, each formula comes with a finite variable context. In order to assign a truth value to the formula, we need an interpretation of its context: an assignment of the variables to elements of a structure. I will describe a categorical generalization of first-order logic, obtained by replacing the category of finite sets (variable contexts) with any small category C with finite colimits, and replacing arbitrary sets (domains of structures) with formal directed colimits from C. I will present a deductive system and completeness theorem for this logic, which is related to hyperdoctrines, a notion from categorical logic. Once this categorical framework is in place, it is easily dualizable. The result is a first-order "cologic", which is well-suited for studying profinite structures in terms of their finite quotients; indeed, this was the original motivation. As particular examples, I will explain how the framework includes the "cologic" of profinite groups due to Cherlin, Macintyre, and van den Dries, and the theories of projective Fraisse limits due to Solecki and Irwin. 

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