Geodesic
A comonad for Grothendieck fibrations
Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 371-389.

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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.
Publié le :
Classification : 18N45, 03G30, 18N10
Keywords: Grothendieck fibration, lax idempotent monad 
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Jacopo Emmenegger; Luca Mesiti; Giuseppe Rosolini; Thomas Streicher. A comonad for Grothendieck fibrations. Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 371-389. http://geodesic.mathdoc.fr/item/TAC_2024_40_a12/