{"paper":{"title":"FI-sets with relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"David Speyer, Eric Ramos, Graham White","submitted_at":"2018-04-11T21:50:21Z","abstract_excerpt":"Let FI denote the category whose objects are the sets $[n] = \\{1,\\ldots, n\\}$, and whose morphisms are injections. We study functors from the category FI into the category of sets. We write $\\mathfrak{S}_n$ for the symmetric group on $[n]$. Our first main result is that, if the functor $[n] \\mapsto X_n$ is \"finitely generated\" there there is a finite sequence of integers $m_i$ and a finite sequence of subgroups $H_i$ of $\\mathfrak{S}_{m_i}$ such that, for $n$ sufficiently large, $X_n \\cong \\bigsqcup_i \\mathfrak{S}_n/(H_i \\times \\mathfrak{S}_{n-m_i})$ as a set with $\\mathfrak{S}_n$ action. Our "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.04238","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}