Bibliographic Information: Henry, S. (2024, October 9). When does Indκ(CI) ≃Indκ(C)I? [arXiv:2307.06664v2]. arXiv. https://arxiv.org/abs/2307.06664v2
Research Objective: This paper investigates the conditions under which the $\kappa$-ind completion of a functor category, denoted $\operatorname{Ind}\kappa(C^I)$, is equivalent to the category of functors from $I$ to the $\kappa$-ind completion of $C$, denoted $\operatorname{Ind}\kappa(C)^I$. This question is closely related to the accessibility of functor categories and the characterization of their locally presentable objects.
Methodology: The paper employs techniques from category theory, particularly those related to ind-completions, accessible categories, and functor categories. It constructs explicit counterexamples to disprove a previous incorrect theorem and provides rigorous proofs for two corrected theorems.
Key Findings:
Main Conclusions: The paper provides a comprehensive analysis of the conditions under which the ind-completion commutes with functor categories. The corrected theorems and counterexamples clarify the relationship between these constructions and have implications for the study of accessible categories and locally presentable categories.
Significance: This research contributes to the understanding of ind-completions and functor categories, fundamental concepts in category theory. The results have implications for various areas where these concepts are used, such as algebraic geometry, topology, and theoretical computer science.
Limitations and Future Research: The paper primarily focuses on ordinary categories. Exploring similar questions in the context of enriched categories or higher categories could be a potential avenue for future research. Additionally, investigating the implications of these results for specific applications of ind-completions and functor categories could lead to further insights.
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by Simon Henry at arxiv.org 10-10-2024
https://arxiv.org/pdf/2307.06664.pdfDeeper Inquiries