In this paper, which is subsequent to our previous paper (but can be read independently from it), we continue our study of the closed model structure on the category of small weakly unital dg categories (in the sense of Kontsevich-Soibelman) over a field. In our previous paper, we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, namely the idempotency $\id_x\cdot \id_x=\id_x$ for any object. Although this condition led us to a great simplification, it was redundant and had to be dropped.
Here we get rid of this condition, and provide a closed model structure in full generality. The new closed model category is as well cofibrantly generated, and it is proven to be Quillen equivalent to the closed model category of (strictly unital) dg categories, given by Tabuada. Dropping the idempotency condition makes the construction of the closed model structure more distant from loc.cit., and requires new constructions. One of them is a pre-triangulated hull of a wu dg category, which in turn is shown to be a wu dg category as well.
One example of a weakly unital dg category which naturally appears is the bar-cobar resolution of a dg category. We supply this paper with a refinement of the classical bar-cobar resolution of a unital dg category which is strictly unital (appendix B). A similar construction can be applied to constructing a cofibrant resolution of weakly unital dg categories.
Keywords: dg-category, closed model category, weak units
2020 MSC: 18N40, 18G35
Theory and Applications of Categories, Vol. 37, 2021, No. 15, pp 388-417.
Published 2021-04-19.
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