Descent for fibred monads

[Richard Garner <richard.garner@mq.edu.au>]

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


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