Cartesian exponentiation and monadicity
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914789552291840 |
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| author | Riehl, Emily Verity, Dominic |
| author_facet | Riehl, Emily Verity, Dominic |
| contents | An important result in quasi-category theory due to Lurie is the that cocartesian fibrations are exponentiable, in the sense that pullback along a cocartesian fibration admits a right Quillen right adjoint that moreover preserves cartesian fibrations; the same is true with the cartesian and cocartesian fibrations interchanged. To explicate this classical result, we prove that the pullback along a cocartesian fibration between quasi-categories forms the oplax colimit of its "straightening," a homotopy coherent diagram valued in quasi-categories, recovering a result first observed by Gepner, Haugseng, and Nikolaus. As an application of the exponentiation operation of a cartesian fibration by a cocartesian one, we use the Yoneda lemma to construct left and right adjoints to the forgetful functor that carries a cartesian fibration over B to its obB-indexed family of fibers, and prove that this forgetful functor is monadic and comonadic. This monadicity is then applied to construct the reflection of a cartesian fibration into a groupoidal cartesian fibration, whose fibers are Kan complexes rather than quasi-categories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_09853 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cartesian exponentiation and monadicity Riehl, Emily Verity, Dominic Category Theory Algebraic Topology 18A30, 18G55, 55U35, 55U40 An important result in quasi-category theory due to Lurie is the that cocartesian fibrations are exponentiable, in the sense that pullback along a cocartesian fibration admits a right Quillen right adjoint that moreover preserves cartesian fibrations; the same is true with the cartesian and cocartesian fibrations interchanged. To explicate this classical result, we prove that the pullback along a cocartesian fibration between quasi-categories forms the oplax colimit of its "straightening," a homotopy coherent diagram valued in quasi-categories, recovering a result first observed by Gepner, Haugseng, and Nikolaus. As an application of the exponentiation operation of a cartesian fibration by a cocartesian one, we use the Yoneda lemma to construct left and right adjoints to the forgetful functor that carries a cartesian fibration over B to its obB-indexed family of fibers, and prove that this forgetful functor is monadic and comonadic. This monadicity is then applied to construct the reflection of a cartesian fibration into a groupoidal cartesian fibration, whose fibers are Kan complexes rather than quasi-categories. |
| title | Cartesian exponentiation and monadicity |
| topic | Category Theory Algebraic Topology 18A30, 18G55, 55U35, 55U40 |
| url | https://arxiv.org/abs/2101.09853 |