{"corpus_id":117486270,"paper_sha":"bdd465314086b160a3ff21ab17206ed2df141546","doi":null,"arxiv_id":"1106.3772","pmid":null,"pmcid":null,"mag_id":1544316656,"dblp_id":null,"acl_id":null,"title":"Cellular Stratified Spaces I: Face Categories and Classifying Spaces","year":2011,"publication_date":"2011-06-19","venue":"","journal":{"name":"arXiv: Algebraic Topology","pages":null,"volume":""},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics"],"reference_count":49,"citation_count":15,"influential_citation_count":0,"is_open_access":false,"arxiv_categories":["math.AT","math.CO","math.GT"],"arxiv_license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":null,"s2_open_access_landing_url":null,"s2_open_access_license":null,"s2_open_access_status":null,"pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, Gonzalez, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally normal regular cellular stratified space $X$ can be embedded in $X$ as a strong deformation retract. \nHere we elaborate on this idea and develop the theory of cellular stratified spaces. We introduce the notion of cylindrically normal cellular stratified spaces and associate a topological category $C(X)$, called the face category, to such a stratified space $X$. We show that the classifying space $BC(X)$ of $C(X)$ can be naturally embedded into $X$. When $X$ is a cell complex, the embedding is a homeomorphism and we obtain an extension of the barycentric subdivision of regular cell complexes. Furthermore, when the cellular stratification on $X$ is locally polyhedral, we show that $BC(X)$ is a deformation retract of $X$. \nWe discuss possible applications at the end of the paper. In particular, the results in this paper can be regarded as a common framework for the Salvetti complex for the complement of a complexified hyperplane arrangement and a version of Morse theory due to Cohen, Jones, and Segal.","claims":[{"public_id":"cl_09241b48b73d2c52368eb4bcec4ac44e","status":"active","text":"A cylindrically normal cellular stratified space admits an associated topological face category whose classifying space can be naturally embedded into the space.","confidence":0.96,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_09241b48b73d2c52368eb4bcec4ac44e"},{"public_id":"cl_51ca8348bcf278bb6071abfa45f24d81","status":"active","text":"For a cell complex, the embedding of the classifying space is a homeomorphism and extends barycentric subdivision of regular cell 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