Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.01474 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910332993142784 |
|---|---|
| author | Emmenegger, Jacopo Mesiti, Luca Rosolini, Giuseppe Streicher, Thomas |
| author_facet | Emmenegger, Jacopo Mesiti, Luca Rosolini, Giuseppe Streicher, Thomas |
| contents | We prove that cloven Grothendieck fibrations over a fixed base $\ct{B}$ are the pseudo-coalgebras for a lax idempotent 2-comonad on $\ct{Cat}/\ct{B}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01474 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A comonad for Grothendieck fibrations Emmenegger, Jacopo Mesiti, Luca Rosolini, Giuseppe Streicher, Thomas Category Theory 18N45, 03G30, 18N10 We prove that cloven Grothendieck fibrations over a fixed base $\ct{B}$ are the pseudo-coalgebras for a lax idempotent 2-comonad on $\ct{Cat}/\ct{B}$. 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. |
| title | A comonad for Grothendieck fibrations |
| topic | Category Theory 18N45, 03G30, 18N10 |
| url | https://arxiv.org/abs/2305.01474 |