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Limits in enhanced 2-categories of models

Investigate limits in enhanced 2-categories of models Mod_{w', w}(S, C), including the existence, construction, creation, and preservation properties of such limits with respect to evaluation functors, and clarify the behavior of (w', w)-morphisms; where appropriate, relate these results to dotted limits and to the general framework of enhanced 2-categories.

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Background

The paper develops a general theory of enhanced 2-sketches and the associated enhanced 2-categories of models Mod_{w', w}(S, C), providing key results for tight limits and a symmetry-of-internalisation theorem. While Appendix A proves that tight limits are (s, w)-created, a comprehensive account of limits in enhanced 2-categories of models for arbitrary weaknesses remains undeveloped.

A deeper theory would systematize when limits exist and how they interact with weak morphisms (lax, colax, pseudo) in Mod_{w', w}(S, C), and potentially connect with the framework of dotted limits. Such a development would extend and clarify the interaction between enriched/2-categorical limit theory and the enhanced setting.

References

An open problem is to obtain a deeper understanding of limits in $$-categories of models and their $(w', w)$-morphisms, which we expect to be closely connected to the main results of . The dotted limits of may be a useful tool for this pursuit.

— 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