Sketches for arithmetic universes

Abstract

A theory of sketches for arithmetic universes (AUs) is developed, as a base-independent surrogate for suitable geometric theories.

A restricted notion of sketch, called here \emph{context}, is defined with the property that every non-strict model is uniquely isomorphic to a strict model. This allows us to reconcile the syntactic, dealt with strictly using universal algebra, with the semantic, in which non-strict models must be considered.

For any context T, a concrete construction is given of the AU AU<T> freely generated by it.

A 2-category Con of contexts is defined, with a full and faithful 2-functor to the 2-category of AUs and strict AU-functors, given by T |-> AU<T>. It has finite pie limits,  and also all pullbacks of a certain class of ``extension'' maps. Every object, morphism or 2-cell of Con is a finite structure.