On continuity of accessible functors

Giacomo Tendas

doi:10.1007/s10485-022-09677-x

On continuity of accessible functors

Giacomo Tendas

School of Mathematical and Physical Sciences

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

43 Downloads (Pure)

Abstract

We prove that for each locally α-presentable category K there exists a regular cardinal γ such that any α-accessible functor out of K (into another locally α-presentable category) is continuous if and only if it preserves γ-small limits; as a consequence we obtain a new adjoint functor theorem specific to the α-accessible functors out of K. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small V-category is accessible if and only if it is Cauchy complete.

Original language	English
Pages (from-to)	937-946
Number of pages	10
Journal	Applied Categorical Structures
Volume	30
Issue number	5
Early online date	31 Mar 2022
DOIs	https://doi.org/10.1007/s10485-022-09677-x
Publication status	Published - Oct 2022

Bibliographical note

Keywords

Adjoint functor theorems
Continuous functors
Enriched categories
Locally presentable categories

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10.1007/s10485-022-09677-xLicence: CC BY

Publisher version (open access)Final published version, 270 KBLicence: CC BY

Cite this

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N2 - We prove that for each locally α-presentable category K there exists a regular cardinal γ such that any α-accessible functor out of K (into another locally α-presentable category) is continuous if and only if it preserves γ-small limits; as a consequence we obtain a new adjoint functor theorem specific to the α-accessible functors out of K. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small V-category is accessible if and only if it is Cauchy complete.

AB - We prove that for each locally α-presentable category K there exists a regular cardinal γ such that any α-accessible functor out of K (into another locally α-presentable category) is continuous if and only if it preserves γ-small limits; as a consequence we obtain a new adjoint functor theorem specific to the α-accessible functors out of K. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small V-category is accessible if and only if it is Cauchy complete.

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KW - Continuous functors

KW - Enriched categories

KW - Locally presentable categories

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ER -

On continuity of accessible functors

Abstract

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Keywords

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