[Paper Review] Extensions of flat functors and theories of presheaf type

Olivia Caramello|arXiv (Cornell University)|Apr 17, 2014

Homotopy and Cohomology in Algebraic Topology18 references3 citations

TL;DR

This paper develops a general theory of extensions of flat functors in topos theory and establishes a characterization theorem that provides necessary and sufficient semantic conditions for a geometric theory to be of presheaf type—meaning its classifying topos is equivalent to a presheaf topos. The key contribution is a unified criterion based on models in Grothendieck toposes, subsuming prior results and enabling practical verification and construction of new theories of presheaf type.

ABSTRACT

We develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a characterization theorem providing necessary and sufficient semantic conditions for a theory to be of presheaf type. This theorem subsumes all the previous partial results obtained on the subject and has several corollaries which can be used in practice for testing whether a given theory is of presheaf type as well as for generating new examples of theories belonging to this class. Along the way, we establish a number of other results of independent interest, including developments about colimits in the context of indexed categories, expansions of geometric theories and methods for constructing theories classified by a given presheaf topos.

Motivation & Objective

• To develop a comprehensive theory of extensions of flat functors along geometric morphisms in topos theory.
• To provide a complete characterization of geometric theories whose classifying topos is a presheaf topos (i.e., theories of presheaf type).
• To unify and generalize prior partial criteria for identifying theories of presheaf type.
• To offer practical tools for testing whether a given theory is of presheaf type and for constructing new examples.
• To establish connections between semantic properties of models and the syntactic structure of geometric theories.

Proposed method

• Introduces and analyzes $oldsymbol{rak{E}}$-indexed colimits in toposes, particularly in the context of internal diagrams and tensor products.
• Develops a general framework for extending flat functors along geometric morphisms, including adjunctions and embeddings of categories.
• Applies the theory to characterize finitely presentable models and homomorphisms in the context of geometric theories.
• Introduces the concept of semantic $rak{E}$-finite presentability and relates it to strong finite presentability in models.
• Uses expansions of geometric theories and injectivizations to generate new theories of presheaf type.
• Applies the characterization theorem to specific examples, including vector spaces, abelian $\ell$-groups with strong unit, and Diers' fields.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient semantic conditions for a geometric theory to be of presheaf type?
• RQ2 How can one determine whether a given geometric theory is classified by a presheaf topos?
• RQ3 What constructions preserve the property of being of presheaf type, and how can new such theories be generated?
• RQ4 In what way do models in Grothendieck toposes reflect the syntactic structure of a theory of presheaf type?
• RQ5 How do expansions and injectivizations of theories relate to the class of theories of presheaf type?

Key findings

• The paper establishes a characterization theorem stating that a geometric theory is of presheaf type if and only if its finitely presentable models satisfy specific semantic conditions related to extensions of flat functors in Grothendieck toposes.
• The theory of abelian $\ell$-groups with strong unit is proven to be of presheaf type, with finitely presentable models precisely those that are finitely presented as $\mathbb{H}$-models and have a strong unit.
• The theory of vector spaces of dimension $\leq n$ over a field $K$ is shown to be of presheaf type, with models in $\mathbf{Set}$ being precisely the finite-dimensional $K$-vector spaces of dimension $\leq n$.
• The theory of Diers’ fields is identified as a theory of presheaf type, with models being fields that are algebraic over a given field.
• The theory of algebraic extensions of a given field is proven to be of presheaf type, with finitely presentable models corresponding to finite algebraic extensions.
• The theory of abelian $\ell$-groups with strong unit is shown to be Morita-equivalent to the theory of MV-algebras, confirming its place within the class of theories of presheaf type.