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.


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