2014/06/30 by Hugo V. Bacard, Bacard, Hugo V.
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #math.AG #math.AT #math.CT #math.KT
paper · pdf · doi:10.48550/arxiv.1406.7666
24 pages. First draft. Comments are always welcome. arXiv admin note: text overlap with arXiv:1406.1115
arxiv created 2014/06/30 · openalex publication_date 2014/06/30 · arxiv updated 2014/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathscrM be a combinatorial and left proper model category, possibly with a monoidal structure. If \mathscrO is either a monad on \mathscrM or an operad enriched over \mathscrM, define a QS-algebra in \mathscrM to be a weak equivalence \mathscrF: s(\mathscrF) \xrightarrow∼t(\mathscrF) such that the target t(\mathscrF) is an \mathscrO-algebra in the usual sense. A classical \mathscrO-algebra is a QS-algebra supported by an isomorphism \mathscrF. A QS-structure \mathscrF is also a weak equivalence such that t(\mathscrF) has a structure, e.g, Hodge, twistorial, schematic, sheaf, etc. We build a homotopy theory of these objects and compare it with that of usual \mathscrO-algebras/structures. Our results rely on Smith's theorem on left Bousfield localization for combinatorial and left proper model categories. These ideas are derived from the theory of co-Segal algebras and categories.