{"paper":{"title":"Kan extensions and the calculus of modules for $\\infty$-categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CT","authors_text":"Dominic Verity, Emily Riehl","submitted_at":"2015-07-06T13:47:37Z","abstract_excerpt":"Various models of $(\\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\\infty$-cosmos. In a generic $\\infty$-cosmos, whose objects we call $\\infty$-categories, we introduce modules (also called profunctors or correspondences) between $\\infty$-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from $A$ to $B$ is an $\\infty$-category equipped with a left action of $A$ and a right action of $B$, in a suitable se"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01460","kind":"arxiv","version":3},"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"}