# Categories turned models: taming the finite

## Hans Schoutens

### The City University of New York

Whereas a category theorist sees mathematics as objects interacting with each other via maps, a model theorist looks instead at their internal structure. So we may think of the former as the sociologues of mathematics and the latter as their psychologues. It is well-known that to a first-order theory we can associate the category of its models, but this produces often a non-natural category, as the maps need to be elementary, and maps rarely are! I will discuss the opposite (Jungian?) perspective: viewing a category as a first-order structure. This yields some unexpected rewards: it allows us to define certain second-order concepts, like finiteness, in a first-order way. I will illustrate this with some examples: sets, modules, topologies, …

Professor Schoutens is a professor of mathematics at the City University of New York, and conducts research in algebraic model theory, commutative algebra, algebraic geometry, rigid analytic geometry and valuation theory.