Abstract
Throughout this article, we fix a group G and a commutative ring \(\Bbbk \). This is an exposition on 2-equivalences between a 2-category of small \(\Bbbk \)-categories with pseudo G-actions and a 2-category of G-graded small \(\Bbbk \)-categories, which generalizes (Asashiba in Appl Categor Sturct 25(8):3278–3296, 2017). This article is a translation of some parts of Ch. 4 and 5 in the book (Asashiba in Categories and representation theory, focused on 2-categorical covering theory, SGC library 155, Saiensu-sha, 2019) in Japanese.
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Notes
Representations of algebras without oriented cycles in their quivers are much simpler than others.
Such as a bicategory of all small categories whose 1-morphisms are bimodules over small categories, the compositions are given by tensor products, and 2-morphisms are bimodule morphisms between bimodules.
I first learned how to draw string diagrams by these movies.
Cones \((x, \pi )\) and \((x', \pi ')\) are called isomorphic if there exists an isomorphism \(f:x' \rightarrow x\) in \({{{\mathscr {C}}}}\), such that \(\pi ' = \pi \circ \Delta (f)\).
For example, if \(R \circ \varprojlim \) has an automorphism \(\gamma \) that is not equal to \({\mathbb {1}}_{R \circ \varprojlim }\), then \(\beta := \varprojlim R^I(\pi ) \bullet \gamma \) is a natural isomorphism \(R \circ \varprojlim \Rightarrow \varprojlim \circ R^I\), but \(\beta _F\) is not a morphism from the cone \((R(\varprojlim F), R(\pi _F))\) to the cone \((\varprojlim R^I(F), \pi _{R^I(F)})\) for some F.
They assume that all X(a) (\(a \in G\)) are autoequivalences, which is stronger than (a\('\)), but follows by Remark 6.2. Note that they defined a lax functor version not colax version.
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Communicated by Javad Asadollahi.
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This work is partially supported by a Grant-in-Aid for Scientific Research (C) 18K03207 from JSPS and by JST CREST Mathematics (15656429).
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Asashiba, H. Cohen–Montgomery Duality for Pseudo-actions of a Group. Bull. Iran. Math. Soc. 47, 767–842 (2021). https://doi.org/10.1007/s41980-020-00413-6
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DOI: https://doi.org/10.1007/s41980-020-00413-6