The de Rham homotopy theory and differential graded category

Abstract

This paper is a generalization of [30]. We develop the de Rham homotopy theory of not necessarily nilpotent spaces. We use two algebraic objects: closed dg-categories and equivariant dg-algebras. We see these two objects correspond in a certain way (Prop.3.3.4, Thm.3.4.5). We prove an equivalence between the homotopy category of schematic homotopy types [22] and a homotopy category of closed dg-categories (Thm.1.0.1). We give a description of homotopy invariants of spaces in terms of minimal models (Thm.1.0.2). The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence between fiberwise rationalizations [5] and closed dg-categories with subsidiary data (Thm.1.0.4).

Tannaka duality and stable infinity-categories

“…The author does not know which formalism is the most adequate one (perhaps it depends on the purposes). If we consider the fine ‐category to be a categorical invariant of , then this viewpoint is similar to the idea by Moriya that the Tannakian differential graded category (in the sense of loc. cit .)…”

“…Remark There are several formalisms of rational homotopy types and rationalizations of non‐nilpotent topological space: for example, fiberwise rationalizations, schematizations, pro‐algebraic homotopy types, Tannakian differential graded categories (see ). The author does not know which formalism is the most adequate one (perhaps it depends on the purposes).…”

“…associated with a topological space is a model of a rationalization of . The main difference is that our construction of may be viewed as a generalization of or in algebraic geometry, while the construction in is a generalization of Sullivan's construction which is a ‘de Rham‐theoretic’ approach. On the other hand, in light of Theorem , we might think that the associated stack is a candidate for a model of a rationalization of .…”

“…There have been various works on rational homotopy theory for non‐nilpotent spaces, for example, see Bousfield and Kan , Brown and Szczarba , Gómez‐Tato et al. , Toën , Pridham , and Moriya (cf. Remark ).…”

Tannaka duality and stable infinity-categories

“…Section 6.5), although we do not obtain a reconstruction of a rational homotopy type from the associated fine ∞-category. There have been various works on rational homotopy theory for non-nilpotent spaces, for example, see Bousfield-Kan [8], Brown-Szczarba [9], Gómez-Tato-Halperin-Tanré [22], Toën [55], Pridham [48], and Moriya [46] (cf. Remark 6.23).…”