Resolutions, higher extensions and the relative Malʼtsev axiom

We study how the concept of higher-dimensional extension which comes from categorical Galois theory relates to simplicial resolutions. For instance, an augmented simplicial object is a resolution if and only if its truncation in every dimension gives a higher extension, in which sense resolutions ar...

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Bibliographic Details
Published in:Journal of algebra 2012-12, Vol.371, p.132-155
Main Authors: Everaert, Tomas, Goedecke, Julia, Van der Linden, Tim
Format: Article
Language:English
Subjects:
Higher extension
Malʼtsev condition
Relative homological algebra
Simplicial resolution
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Summary:We study how the concept of higher-dimensional extension which comes from categorical Galois theory relates to simplicial resolutions. For instance, an augmented simplicial object is a resolution if and only if its truncation in every dimension gives a higher extension, in which sense resolutions are infinite-dimensional extensions or higher extensions are finite-dimensional resolutions. We also relate certain stability conditions of extensions to the Kan property for simplicial objects. This gives a new proof of the fact that a regular category is Malʼtsev if and only if every simplicial object is Kan, using a relative setting of extensions.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2012.07.036