A comonad for Grothendieck fibrations

Joint work with Emmenegger, Rosolini, 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.

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