{"paper":{"title":"CW-complexes in the Category of Small Categories","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.AT","math.KT"],"primary_cat":"math.CT","authors_text":"Andrew Salch, Christian Frank","submitted_at":"2017-11-23T05:25:45Z","abstract_excerpt":"We compute the collection of CW-complexes in the model category of small categories constructed by Joyal and Tierney. More generally, if $X$ is a connected topological space, we show that the homotopy category of CW-complexes in Joyal-Tierney's model category of sheaves of sets on $X$ is equivalent to the homotopy category of groupoids. As an application of the ideas, we show that the algebraic $K$-theory groups of the category of pointed small categories are trivial, and more generally, the algebraic $K$-theory groups of any sufficiently \"nice\" Waldhausen category $\\mathcal{A}$ of pointed sma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08579","kind":"arxiv","version":1},"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"}