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Accessibility and (co)limits for models of PIE-limit 2-theories

Prove the conjecture that for any appropriately nice PIE-limit 2-theory, the 2-category whose objects are models and whose 1-cells are pseudo morphisms is accessible as a 2-category and admits the specific limits and colimits described in the cited framework for PIE-limit 2-theories.

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Background

The paper positions enhanced 2-sketches as a framework that subsumes PIE-limit 2-theories and refines their morphisms of models. Prior work conjectured that for well-behaved PIE-limit 2-theories the 2-category of models and pseudomorphisms is accessible and supports certain limits and colimits, but a proof was deferred pending development of a suitable theory of limit enhanced 2-theories.

Establishing this result would provide structural guarantees (accessibility and the existence of specified (co)limits) for 2-categories of models in the PIE-limit setting, aligning it with known one-dimensional presentability theory and supporting broader applications.

References

Along similar lines, in Theorem~9.4, it was conjectured that the 2-category of models and pseudo morphisms of any appropriately nice PIE-limit 2-theory would be accessible as a 2-category, and admit certain limits and colimits (cf. Definitions~4.1 {content} 4.5). The proof of this conjecture was deferred until a theory of limit $$-theories had been developed.

— Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation (2412.07475 - Arkor et al., 10 Dec 2024) in Section 9, Future directions