Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics
Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday
July 18-21 2005, Australian National University, Canberra
Wed, 20 July: 16:15 - 16:55
Descent on 2-fibrations and strongly 2-regular 2-categories
Hermida, Claudio (Instituto Superior Técnico, Lisbon)
We consider pseudo-descent in the context of 2-fibrations.
A 2-category of descent data is associated to a 3-truncated
simplicial object in the base 2-category.
A morphism $q$ in the base induces (via comma-objects and pullbacks)
an internal category whose truncated simplicial nerve induces in
turn the 2-category of descent data for $q$. When the 2-fibration admits direct
images, we provide the analogues of the Beck--Bénabou--Roubaud
theorem, identifying the 2-category of descent data with that of
pseudo-algebras for the pseudo-monad $q^{*}\Sigma_{q}$.
We introduce a notion of \textit{strong 2-regularity\/} for a 2-category ${\mathcal R}$, so that its basic 2-fibration of internal fibrations
${\mathit cod}:{\mathsf{Fib}}({\mathcal R})\rightarrow{\mathcal R}$ admits
direct images. In this context, we show that \textit{essentially-surjective-on-objects\/} morphisms,
defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.