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 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 . 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.
Highlights
My 1999 paper “Topical Categories of Domains” [12], which used geometric logic to deal with toposes as generalized spaces, makes the following comment on the logic.It seems to us that in the work of the paper the infinities are restricted to those that can be accessed effectively through free algebra constructions
This emboldens us to hope that the full geometric logic is unnecessary, that it suffices to have coherent logic with assorted free algebras, and that [Grothendieck toposes] could be replaced by Joyal’s arithmetic universes [AUs]
It is analogous to the 2-category Top of Grothendieck toposes and geometric morphisms, or, more generally, to BTop/S, where S is a chosen base elementary topos with nno, and BTop/S is the 2-category of bounded S -toposes
Summary
My 1999 paper “Topical Categories of Domains” [12], which used geometric logic to deal with toposes as generalized spaces, makes the following comment on the logic. It seems to us that in the work of the paper the infinities are restricted to those that can be accessed effectively through free algebra constructions This emboldens us to hope that the full geometric logic is unnecessary, that it suffices to have coherent logic with assorted free algebras, and that [Grothendieck toposes] could be replaced by Joyal’s arithmetic universes [AUs]. Con is based on a semantics using arithmetic universes, its objects are not AUs as such, but finite structures, presentations of AUs in the style of sketches. It is as if we defined Grothendieck toposes to be the geometric theories they classify, with no attempt to identify equivalent presentations. We finish with a proof that the entire construction can be internalized in any AU, as anticipated by Joyal’s original work on Godel’s Theorem
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