#
A comonad for Grothendieck fibrations

##
Jacopo Emmenegger, Luca Mesiti,
Giuseppe Rosolini, and Thomas Streicher

We prove that cloven Grothendieck fibrations over a fixed base B are the
pseudo-coalgebras for a lax idempotent 2-comonad on Cat/B. We show this via an
original observation that the known colax idempotent 2-monad for fibrations over a fixed
base has a right 2-adjoint. As an important consequence, we obtain an original cofree
construction of a fibration on a functor. We also give a new, conceptual proof of the fact
that the forgetful 2-functor from split fibrations to cloven fibrations over a fixed base
has both a left 2-adjoint and a right 2-adjoint, in terms of coherence phenomena of
strictification of pseudo-(co)algebras. The 2-monad for fibrations yields the left
splitting and the 2-comonad yields the right splitting. Moreover, we show that the
constructions induced by these coherence theorems recover Giraud's explicit
constructions of the left and the right splittings.

Keywords: Grothendieck fibration, lax idempotent monad

2020 MSC:
18N45, 03G30, 18N10

*Theory and Applications of Categories,*
Vol. 40, 2024,
No. 13, pp 371-389.

Published 2024-05-07.

http://www.tac.mta.ca/tac/volumes/40/13/40-13.pdf

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