Quillen-Segal objects and structures: an overview

Abstract

Let $\mathcal{M}$ be a combinatorial and left proper model category, possibly with a monoidal structure. If $\mathcal{O}$ is either a monad on $\mathcal{M}$ or an operad enriched over $\mathcal{M}$, define a QS-algebra in $\mathcal{M}$ to be a weak equivalence $\mathcal{F}:s(\mathcal{F})\stackrel{\sim }{\to }t(\mathcal{F})$ such that the target $t(\mathcal{F})$ is an $\mathcal{O}$-algebra in the usual sense. A classical $\mathcal{O}$-algebra is a QS-algebra supported by an isomorphism $\mathcal{F}$. A QS-structure $\mathcal{F}$ is also a weak equivalence such that $t(\mathcal{F})$ 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 $\mathcal{O}$-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.

24 pages. First draft. Comments are always welcome. arXiv admin note: text overlap with arXiv:1406.1115

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