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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.00607 |
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| _version_ | 1866910627031678976 |
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| author | Bergfalk, Jeffrey |
| author_facet | Bergfalk, Jeffrey |
| contents | The following is an introduction to the study of higher walks, by which we mean a family of higher-dimensional extensions of Todorcevic's method of walks on the ordinals. After a brief review of this method, including, for example, definitions of the classical functions $\mathrm{Tr}$ and $ρ_2$ induced by a choice of $C$-sequence, we record a shortlist of desiderata for such extensions, along with $(n+1)$-dimensional functions $\mathrm{Tr}_n$ and $ρ_2^n$ (induced by a choice of higher-dimensional $C$-sequence) which we show to satisfy the bulk of them. Much of the interest of these higher walks functions lies in their affinity, as in the classical $n=1$ case, for the ordinals $ω_n$ (we show, for example, that $ρ^n_2$ determines both $n$-dimensional linear orderings and $n$-coherent families on $ω_n$, and that higher walks define nontrivial elements of the $n^{\mathrm{th}}$ cohomology groups of $ω_n$), and in the questions that they thereby raise both about the combinatorics of the latter and about higher-dimensional infinitary combinatorics more generally; we collect the most prominent of these questions in our conclusion. These objects are also, though, of a sufficient combinatorial richness to be of interest in their own right, as we have underscored via an extended study of the first genuine novelty among them, the function $\mathrm{Tr}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00607 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An introduction to higher walks Bergfalk, Jeffrey Logic 03E10, 03E05 The following is an introduction to the study of higher walks, by which we mean a family of higher-dimensional extensions of Todorcevic's method of walks on the ordinals. After a brief review of this method, including, for example, definitions of the classical functions $\mathrm{Tr}$ and $ρ_2$ induced by a choice of $C$-sequence, we record a shortlist of desiderata for such extensions, along with $(n+1)$-dimensional functions $\mathrm{Tr}_n$ and $ρ_2^n$ (induced by a choice of higher-dimensional $C$-sequence) which we show to satisfy the bulk of them. Much of the interest of these higher walks functions lies in their affinity, as in the classical $n=1$ case, for the ordinals $ω_n$ (we show, for example, that $ρ^n_2$ determines both $n$-dimensional linear orderings and $n$-coherent families on $ω_n$, and that higher walks define nontrivial elements of the $n^{\mathrm{th}}$ cohomology groups of $ω_n$), and in the questions that they thereby raise both about the combinatorics of the latter and about higher-dimensional infinitary combinatorics more generally; we collect the most prominent of these questions in our conclusion. These objects are also, though, of a sufficient combinatorial richness to be of interest in their own right, as we have underscored via an extended study of the first genuine novelty among them, the function $\mathrm{Tr}_2$. |
| title | An introduction to higher walks |
| topic | Logic 03E10, 03E05 |
| url | https://arxiv.org/abs/2410.00607 |