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Lax comma 2-categories and admissible 2-functors

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Maria Manuel Clementino and Fernando Lucatelli Nunes

This paper is a contribution towards a two dimensional extension of the basic
ideas and results of Janelidze's Galois theory. In the present paper, we give a suitable counterpart notion
to that of *absolute admissible Galois structure* for the lax idempotent context, compatible with
the context of *lax orthogonal factorization systems*. As part of this work, we study lax comma 2-categories,
giving analogue results to the basic properties of the usual comma categories. We show that each morphism of a 2-category induces a 2-adjunction between lax comma 2-categories and comma 2-categories, playing the role of the usual *change-of-base functors*. With these induced 2-adjunctions, we are able to show that each 2-adjunction induces 2-adjunctions between lax comma 2-categories and comma 2categories, which are our analogues of the usual lifting to the comma categories used in Janelidze's Galois theory. We give sufficient conditions under which these liftings are 2-premonadic and induce a lax idempotent 2-monad, which corresponds to our notion of 2-admissible 2-functor. In order to carry out this work, we analyse when a composition of
2-adjunctions is a lax idempotent 2-monad, and when it is 2-premonadic. We give then examples of our 2-admissible 2-functors
(and, in particular, simple 2-functors), especially using a result that says that all admissible (2-)functors in the
classical sense are also 2-admissible (and hence simple as well).

Keywords:
change-of-base functor, comma object, Galois theory,
Kock-Zöberlein monads, semi-left exact functor,
lax comma 2-categories, simple 2-adjunctions, 2-admissible 2-functor

2020 MSC:
18N10, 18N15, 18A05, 18A22, 18A40

*Theory and Applications of Categories,*
Vol. 40, 2024,
No. 6, pp 180-226.

Published 2024-04-05.

http://www.tac.mta.ca/tac/volumes/40/6/40-06.pdf

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