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 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.
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.
| Original language | English |
|---|---|
| Number of pages | 55 |
| Journal | Journal of Logic and Analysis |
| Volume | 11 |
| DOIs | |
| Publication status | Published - Jun 2019 |