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When is the Ind-Completion of a Functor Category Equivalent to the Functor Category of the Ind-Completion?


Core Concepts
This paper investigates when 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 $\operatorname{Ind}\kappa(C)^I$. While a previous theorem incorrectly claimed this holds for any Cauchy complete category $C$ and $\kappa$-small category $I$, this paper disproves that claim and provides two corrected theorems. The first corrected theorem states the equivalence holds if $C$ has $\kappa$-small colimits and $I$ is $\kappa$-small. The second corrected theorem states the equivalence holds if $C$ is an arbitrary category and $I$ is well-founded and $\kappa$-small. The paper provides counterexamples and discusses the optimality of these conditions.
Abstract
  • 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:

    • The paper disproves the claim that $\operatorname{Ind}\kappa(C^I) \simeq \operatorname{Ind}\kappa(C)^I$ holds for any Cauchy complete category $C$ and $\kappa$-small category $I$.
    • It proves that the equivalence holds if $C$ has $\kappa$-small colimits and $I$ is $\kappa$-small.
    • It also proves that the equivalence holds if $C$ is an arbitrary category and $I$ is well-founded and $\kappa$-small.
    • The paper demonstrates the optimality of these conditions, meaning the equivalence holds for all $C$ if and only if $I$ satisfies the given assumptions.
  • 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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Deeper Inquiries

How do these results about ind-completions of functor categories generalize to other categorical constructions, such as comma categories or Grothendieck constructions?

Investigating the generalization of these results to other categorical constructions like comma categories or Grothendieck constructions is a fruitful avenue for further research. Here's a breakdown of potential approaches and challenges: Comma Categories: Challenges: Comma categories $(F \downarrow G)$ depend on the interplay of two functors, $F$ and $G$, making the analysis more intricate. The existence and properties of colimits in $(F \downarrow G)$ are less straightforward compared to functor categories. Potential Approaches: One could start by considering specific cases: Slice Categories: Investigate when $\operatorname{Ind}\kappa(C/X) \simeq \operatorname{Ind}\kappa(C)/X$ for a category $C$ and object $X$. This is a special case of comma categories where one functor is constant. Functors with Adjoints: If $F$ or $G$ has an adjoint, the comma category inherits additional structure, potentially simplifying the analysis of ind-completions. Grothendieck Constructions: Challenges: Grothendieck constructions build categories from indexed families of categories. The interaction between the indexing and the ind-completion is non-trivial. Potential Approaches: Fiberwise Analysis: Examine if the equivalence holds fiberwise, i.e., if $\operatorname{Ind}_\kappa(C_i) \simeq D_i$ for all $i$ in the indexing category, does this imply an equivalence for the Grothendieck constructions? Restrictions on Indexing Categories: Explore if restrictions on the indexing category (e.g., being well-founded, having specific colimit properties) lead to desirable results. General Remarks: Preservation of Colimits: A key aspect of the results about functor categories is the preservation of $\kappa$-filtered colimits by the evaluation functors. Generalizing this to other constructions would require a careful analysis of how colimits are computed in those constructions. Accessibility Rank: The notion of accessibility rank and its interplay with the size of the indexing categories played a crucial role in the original results. Similar considerations would be essential for generalizations.

Could there be a weaker condition on the category $C$ than having $\kappa$-small colimits that still ensures the equivalence $\operatorname{Ind}\kappa(C^I) \simeq \operatorname{Ind}\kappa(C)^I$ holds for all $\kappa$-small categories $I$?

While Theorem 1.2 establishes the equivalence of $\operatorname{Ind}\kappa(C^I) \simeq \operatorname{Ind}\kappa(C)^I$ for categories $C$ with $\kappa$-small colimits and $\kappa$-small categories $I$, finding weaker conditions on $C$ is an interesting open question. Positelski's Theorem: As highlighted in the context, Positelski's Theorem (Theorem 1.5) provides a significant improvement by relaxing the requirement on $C$ to having colimits of $\lambda$-indexed chains for some $\lambda < \kappa$. This suggests that further refinements might be possible. Targeted Colimits: Instead of requiring $C$ to have all $\kappa$-small colimits, one could explore if the existence of specific types of colimits in $C$ (e.g., pushouts, coequalizers) is sufficient to ensure the equivalence for certain classes of $\kappa$-small categories $I$. Counterexamples as Guidance: Analyzing the counterexamples where the equivalence fails when $C$ does not have $\kappa$-small colimits could provide insights into the essential properties needed for the equivalence to hold.

What are the implications of these findings for the study of presheaf categories and their relationship to ind-completions?

These findings have important implications for understanding presheaf categories and their connection to ind-completions: Presheaves as Ind-Completions: Presheaf categories are fundamental in category theory and appear in various contexts. The results highlight that for a small category $C$, the presheaf category $\operatorname{Set}^{C^{op}}$ can be viewed as the ind-completion of $C$ (when considering $\kappa$ larger than the cardinality of $C$). Accessibility and Presheaves: The theorems provide conditions under which functor categories of accessible categories remain accessible. Since presheaf categories are accessible, this sheds light on the structure of categories arising as functor categories from presheaf categories. Representing Functors: The equivalence $\operatorname{Ind}\kappa(C^I) \simeq \operatorname{Ind}\kappa(C)^I$ implies that under suitable conditions, every functor $I \to \operatorname{Ind}_\kappa(C)$ can be represented (up to equivalence) by a functor $I \to C$. This has implications for understanding the representability of functors in terms of simpler objects. Well-Foundedness and Size Issues: The counterexamples and the role of well-foundedness emphasize the subtle interplay between size issues (cardinality of $I$) and the structure of $I$ when considering ind-completions of functor categories. This suggests that careful attention to these aspects is crucial when working with presheaves and ind-completions.
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