Re: Descent for fibred monads

["George Janelidze" <janelg@telkomsa.net>]

Dear Richard,

I would like to see your question formulated more precisely, and showing
general (admissible) Galois theory example instead of the locally connected
topos example. Some days ago you recommended Carboni-Janelidze-Kelly-Pare
paper as one of references for factorization systems (thank you for that!),
and now please look at Section 5 of that paper. Note that
"admissible"="semi-left-exact" can be replaced with "fibration".

Best regards,
George Janelidze

--------------------------------------------------
From: "Richard Garner" <richard.garner@mq.edu.au>
Sent: Thursday, May 15, 2014 1:15 PM
To: "Categories list" <categories@mta.ca>
Subject: categories: Descent for fibred monads

> Dear categorists,
>
> Does the following variant of the Benabou-Roubaud/Beck monadic descent
> theorem appear anywhere?
>
> Let p:E--->B be a fibration with sums and let T:E--->E be a fibred monad
> over B. Let q: E^T ----> B be the induced fibration of T-algebras. Let
> f: x--->y in B. Then to give T-algebra descent data for f---that is, a
> diagram over the kernel-pair of f valued in E^T---is equally to give an
> algebra for the composite monad
>
> E_x ----f_!----> E_y ----T_y---> E_y ---f^*----> E_x
>
> This doesn't seem to be an application of the usual monadic descent
> theorem to q: E^T ---> B; that would identify T-algebra descent data for
> f with algebras for a monad on (E^T)_x, not on E_x.
>
> For example, take E ----> S a connected topos with pi_0 -| Delta -|
> Gamma. Let T be the monad for constant objects on E induced by the
> fibred adjunction pi_0 -| Delta. Given f: U --->> 1 in E, to give
> T-algebra descent data for f is to give a locally constant object split
> by U. So such objects are equally the algebras for the monad
>
> E/U -----> E/U
> (A--->U) |----> (Delta pi_0 A) x U ----> U
>
> In the same situation, take T to be the monad for free vector spaces E
> ---pi_0---> S ---Fv---> S ---Delta---> E induced by the free vector
> space monad Fv on S. Then T-algebra descent data over U --->> 1 is a
> vector bundle split by U; so such objects are equally algebras for the
> monad (A--->U) |----> (Delta Fv pi_0 A) x U ---> U
>
> Richard

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]