Superalgebras and Superstructures: an overview

Hugo Bacard (1)

1 Department of Mathematics [London, Ontario]

Let M be a combinatorial and left proper model category, possibly with a monoidal structure. If O is either a monad on M or an operad enriched over M, define a superalgebra in M to be a weak equivalence F : s(F) → t(F) such that the target t(F) is an O-algebra in the usual sense. A classical O-algebra is a superalgebra supported by an isomorphism F. A superstructure F is also a weak equivalence such that t(F) 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 O-algebras/structures. Our results rely on Smith's theorem on left Bousfield localization for combinatorial and left proper model categories.

Mots clés: Algebras; Hodge structures; model categories; homotopy theory; higher categories

Domaines: Catégories et ensembles [math.CT]; Algèbre commutative [math.AC]; Géométrie algébrique [math.AG]; K-théorie et homologie [math.KT]; Topologie algébrique [math.AT]

Hugo Bacard. Superalgebras and Superstructures: an overview. 2014. ⟨hal-01016219⟩