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

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Piergiorgio Panero and Boris Shoikhet

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.

Keywords:
dg-category, closed model category, weak units

2020 MSC:
18N40, 18G35

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 5, pp 135-155.

Published 2022-01-13.

http://www.tac.mta.ca/tac/volumes/38/5/38-05.pdf

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