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Pseudo-Kan extensions and descent theory

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Fernando Lucatelli Nunes

There are two main constructions in classical descent theory: the category
of algebras and the descent category, which are known to be examples of
weighted bilimits. We give a formal approach to descent theory, employing
formal consequences of commuting properties of bilimits to prove classical
and new theorems in the context of Janelidze-Tholen ``Facets of Descent
II'', such as Benabou-Roubaud Theorems, a Galois Theorem, embedding
results and formal ways of getting effective descent morphisms. In order
to do this, we develop the formal part of the theory on commuting bilimits
via pseudomonad theory, studying idempotent pseudomonads and proving a
2-dimensional version of the adjoint triangle theorem. Also, we work out
the concept of pointwise pseudo-Kan extension, used as a framework to talk
about bilimits, commutativity and the descent object. As a subproduct,
this formal approach can be an alternative perspective/guiding template
for the development of higher descent theory.

Keywords:
descent objects, descent category, Kan extensions, pseudomonads,
biadjunctions, (effective) descent morphism, weighted bilimits,
Benabou-Roubaud Theorem, Galois Theory, commutativity of bilimits

2010 MSC:
18A30, 18A40, 18C15, 18C20, 18D05

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 15, pp 390-444.

Published 2018-05-16.

http://www.tac.mta.ca/tac/volumes/33/15/33-15.pdf

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