The closed model structure on the category of weakly unital dg categories: an addendum

Piergiorgio Panero ; Boris Shoikhet

Theory and applications of categories, Tome 38 (2022), pp. 135-155.

Résumé

This paper is an addendum to our paper [PS], where a closed model structure on the category C_{\dgwu}(k) of small (Kontsevich-Soibelman) weakly unital dg categories is constructed, k a field of any characteristic. In [PS], we referred to our earlier preprint for proofs of the following results: (A) small (co)completeness of C_{\dgwu}(k), and (B) the non-symmetric dg operad O' (which governs weakly unital dg categories acting on k-quivers, we recall its definition in Section 2.12) is quasi-isomorphic to the operad of unital associative algebras Assoc_+, under a natural projection. Recall that the (co)completeness in (A) is the first axiom of a closed model category, while (B) was crucial in the proof in [PS, Th. 5.3] of the Quillen equivalence between C_{\dgwu}(k) (equipped with our model structure) and the category of small dg categories C_{\dgwu}(k) (equipped with the Tabuada model structure for which the weak equivalences are quasi-equivalences). In this paper we collect our earlier proofs of (A) and (B), which serves as an addendum to [PS], and makes these two papers self-contained.

Détail

Publié le : 2022-01-13

Classification : 18N40, 18G35
Keywords: dg-category, closed model category, weak units