All Segal objects are generalised monads in spans
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915640643682304 |
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| author | Kern, David |
| author_facet | Kern, David |
| contents | We extend Barwick's and Haugseng's construction of the double $\infty$-category of spans in a pullback-complete $\infty$-category $\mathfrak{C}$ to more general shapes: for a large class of algebraic patterns $\mathfrak{P}$, we define a $\mathfrak{P}$-monoidal $\infty$-category of $\mathfrak{P}$-shaped spans in $\mathfrak{C}$, and we identify $\mathfrak{P}$-monads in it with Segal $\mathfrak{P}$-objects in $\mathfrak{C}$. For the cell pattern $Θ^{\mathrm{op}}$, this recovers a homotopical reformulation of Batanin's original definition of weak $ω$-categories, and in general can be seen as a variant of the generalised multicategories of Burroni, Hermida, Leinster and Cruttwell-Shulman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | All Segal objects are generalised monads in spans Kern, David Category Theory Algebraic Topology 18N65, 18N70 We extend Barwick's and Haugseng's construction of the double $\infty$-category of spans in a pullback-complete $\infty$-category $\mathfrak{C}$ to more general shapes: for a large class of algebraic patterns $\mathfrak{P}$, we define a $\mathfrak{P}$-monoidal $\infty$-category of $\mathfrak{P}$-shaped spans in $\mathfrak{C}$, and we identify $\mathfrak{P}$-monads in it with Segal $\mathfrak{P}$-objects in $\mathfrak{C}$. For the cell pattern $Θ^{\mathrm{op}}$, this recovers a homotopical reformulation of Batanin's original definition of weak $ω$-categories, and in general can be seen as a variant of the generalised multicategories of Burroni, Hermida, Leinster and Cruttwell-Shulman. |
| title | All Segal objects are generalised monads in spans |
| topic | Category Theory Algebraic Topology 18N65, 18N70 |
| url | https://arxiv.org/abs/2401.04704 |