A Model Structure on Categories Related to the Categories of Complexes
- PDF / 214,046 Bytes
- 17 Pages / 594 x 792 pts Page_size
- 62 Downloads / 167 Views
A MODEL STRUCTURE ON CATEGORIES RELATED TO THE CATEGORIES OF COMPLEXES V. V. Lyubashenko
UDC 512.58
We prove a Hinich-type theorem on the existence of a model structure on a category related by adjunction to the category of differential graded modules over a graded commutative ring.
1. Introduction In [4], Hinich proved the theorem on existence of a model structure on a category related by adjunction to the category of complexes. In the present paper, we give a detailed proof of a similar theorem. The indicated two theorems differ by at least two factors. First, Hinich used dg -modules over a (commutative) ring, whereas we consider differential graded modules over a graded commutative ring k. Second, in his proof, Hinich introduced certain morphisms called elementary trivial cofibrations and showed that any trivial cofibration is a retract of a countable composition of elementary cofibrations. At the same time, we demonstrate that a trivial cofibration is a retract of an elementary trivial cofibrations in our sense. We use our theorem to prove that the categories of bimodules or polymodules over nonsymmetric operads have a model structure [5, 6]. A model structure for modules over operads was constructed by Harper in [3] (Theorem 1.7). After the publication of Hinich’s work [4], numerous results were obtained. Thus, on the basis of a given (monoidal) model category, a model structure was constructed for another category related to the first category by adjunction [1] (Sec. 2.5), on a category of monoids [9] (Theorem 3.1), or on a category of operads [8] (Remark 2), [7] (Theorem 1.1). It is clear that, in this approach, it is necessary first to consider the model category. A category of differential (unbounded) graded k0 -modules has a projective model structure for a commutative ring k0 [2]. As in [1], the same result for a graded commutative ring k can be derived from the case of a commutative ring k0 . After this, it is necessary to prove that dg -k-mod is a monoidal model category, which requires detailed information about cofibrations. The required information is given, e.g., by the proof of a Hinich-type theorem according to which any cofibration is a retract of a countable composition of elementary cofibrations (of a specific form). Thus, for any approach, it is unlikely to be able to avoid technical work. As one more reason for the application of Hinich’s approach, we can mention pedagogic arguments: This approach can be explained to students in detail and by means of examples. Notation and Conventions. In the present paper, the word “graded” means “Z-graded”. Let k be a graded commutative ring (equipped with zero differential). By gr = grk = gr -k-mod we denote a closed category of Z-graded k-modules with k-linear homomorphisms of degree 0. Thus, an object of gr is X = (X m )m2Z . Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine; e-mail: [email protected]. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 232–244, February, 2020. Original article submit
Data Loading...
