Sketches for arithmetic universes
  A theory of sketches for arithmetic universes (AUs) is developed.   A restricted notion of sketch, called here "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. 
Steven Vickers
math.CT