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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.06664 |
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| _version_ | 1866912063358500864 |
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| author | Henry, Simon |
| author_facet | Henry, Simon |
| contents | We investigate under which condition the $κ$-ind completion of a functor category $C^I$ is equivalent to the category of functors from $I$ to the $κ$-ind completion of $C$. A published theorem implies this is true for any Cauchy complete category $C$ and $κ$-small category $I$, but we show this is not the case in general. We prove two results that seem to cover all applications of this incorrect theorem we could find in the literature: The result holds if $C$ has $κ$-small colimits and $I$ is $κ$-small, or if $C$ is an arbitrary category and $I$ is well-founded and $κ$-small. In both cases, we show that the conditions are optimal in the sense that the result holds for all $C$ if and only if $I$ satisfies the given assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06664 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | When does $\operatorname{Ind}_κ(C^I) \simeq \operatorname{Ind}_κ(C)^I$? Henry, Simon Category Theory 18A25, 18C35 We investigate under which condition the $κ$-ind completion of a functor category $C^I$ is equivalent to the category of functors from $I$ to the $κ$-ind completion of $C$. A published theorem implies this is true for any Cauchy complete category $C$ and $κ$-small category $I$, but we show this is not the case in general. We prove two results that seem to cover all applications of this incorrect theorem we could find in the literature: The result holds if $C$ has $κ$-small colimits and $I$ is $κ$-small, or if $C$ is an arbitrary category and $I$ is well-founded and $κ$-small. In both cases, we show that the conditions are optimal in the sense that the result holds for all $C$ if and only if $I$ satisfies the given assumption. |
| title | When does $\operatorname{Ind}_κ(C^I) \simeq \operatorname{Ind}_κ(C)^I$? |
| topic | Category Theory 18A25, 18C35 |
| url | https://arxiv.org/abs/2307.06664 |