Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product
We define a commutative monoid structure on the poset of left-exact localizations of a higher topos, that we call the acyclic product. Our approach is anchored in a structural analogy between the poset of left-exact localizations of a topos and the poset of ideals of a commutative ring. The acyclic...
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| creator | Anel, Mathieu Biedermann, Georg Finster, Eric Joyal, André |
| description | We define a commutative monoid structure on the poset of left-exact
localizations of a higher topos, that we call the acyclic product. Our approach
is anchored in a structural analogy between the poset of left-exact
localizations of a topos and the poset of ideals of a commutative ring. The
acyclic product is analogous to the product of ideals. The sequence of powers
of a given left-exact localization defines a tower of localizations. We show
how this recovers the towers of Goodwillie calculus in the unstable homotopical
setting. We use this to describe the topoi of $n$-excisive functors as
classifying $n$-nilpotent objects. |
| doi_str_mv | 10.48550/arxiv.2308.15573 |
| format | Article |
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localizations of a higher topos, that we call the acyclic product. Our approach
is anchored in a structural analogy between the poset of left-exact
localizations of a topos and the poset of ideals of a commutative ring. The
acyclic product is analogous to the product of ideals. The sequence of powers
of a given left-exact localization defines a tower of localizations. We show
how this recovers the towers of Goodwillie calculus in the unstable homotopical
setting. We use this to describe the topoi of $n$-excisive functors as
classifying $n$-nilpotent objects.</description><identifier>DOI: 10.48550/arxiv.2308.15573</identifier><language>eng</language><subject>Mathematics - Algebraic Topology ; Mathematics - Category Theory</subject><creationdate>2023-08</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,777,882</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2308.15573$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2308.15573$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Anel, Mathieu</creatorcontrib><creatorcontrib>Biedermann, Georg</creatorcontrib><creatorcontrib>Finster, Eric</creatorcontrib><creatorcontrib>Joyal, André</creatorcontrib><title>Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product</title><description>We define a commutative monoid structure on the poset of left-exact
localizations of a higher topos, that we call the acyclic product. Our approach
is anchored in a structural analogy between the poset of left-exact
localizations of a topos and the poset of ideals of a commutative ring. The
acyclic product is analogous to the product of ideals. The sequence of powers
of a given left-exact localization defines a tower of localizations. We show
how this recovers the towers of Goodwillie calculus in the unstable homotopical
setting. We use this to describe the topoi of $n$-excisive functors as
classifying $n$-nilpotent objects.</description><subject>Mathematics - Algebraic Topology</subject><subject>Mathematics - Category Theory</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotz7tOwzAUgGEvDKjwAEx46Opg-8SXsFURl0iRYMiIFJ34IiyFukoNanh6RGH6t1_6CLkRvKqtUvwOl1P6qiRwWwmlDFyStg-xsHBCV2ifHc7pG0vK-yPNkW7f0j6WdcuGfMiJdl13T4f3QHdudXNy9HXJ_tOVK3IRcT6G6_9uyPD4MLTPrH956tpdz1AbYEJ5hUEYqD1ILqSXHANMVlkPvFbop0abgA60Vz5IrZsJZTSyqQEn4SxsyO3f9qwYD0v6wGUdfzXjWQM_HXJDbw</recordid><startdate>20230829</startdate><enddate>20230829</enddate><creator>Anel, Mathieu</creator><creator>Biedermann, Georg</creator><creator>Finster, Eric</creator><creator>Joyal, André</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20230829</creationdate><title>Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product</title><author>Anel, Mathieu ; Biedermann, Georg ; Finster, Eric ; Joyal, André</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a673-15d5ae1734d32012d20ae3b858d3045adb967eac36d5de2669ba2f72943ab1c83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Mathematics - Algebraic Topology</topic><topic>Mathematics - Category Theory</topic><toplevel>online_resources</toplevel><creatorcontrib>Anel, Mathieu</creatorcontrib><creatorcontrib>Biedermann, Georg</creatorcontrib><creatorcontrib>Finster, Eric</creatorcontrib><creatorcontrib>Joyal, André</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Anel, Mathieu</au><au>Biedermann, Georg</au><au>Finster, Eric</au><au>Joyal, André</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product</atitle><date>2023-08-29</date><risdate>2023</risdate><abstract>We define a commutative monoid structure on the poset of left-exact
localizations of a higher topos, that we call the acyclic product. Our approach
is anchored in a structural analogy between the poset of left-exact
localizations of a topos and the poset of ideals of a commutative ring. The
acyclic product is analogous to the product of ideals. The sequence of powers
of a given left-exact localization defines a tower of localizations. We show
how this recovers the towers of Goodwillie calculus in the unstable homotopical
setting. We use this to describe the topoi of $n$-excisive functors as
classifying $n$-nilpotent objects.</abstract><doi>10.48550/arxiv.2308.15573</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Algebraic Topology Mathematics - Category Theory |
| title | Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product |
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