Timeline for "Sameness" of dg and A-infinity categories
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jun 22, 2021 at 15:51 | comment | added | Dmitri Pavlov | @FrancescoGenovese: The paper you referenced is talking about a very different category, the category of A_∞-categories and A_∞-functors (not ordinary enriched functors) between them. As far as I am aware, we don't know how to define A_∞-functors for arbitrary A_∞-operads in arbitrary monoidal model categories, which is the context of my answer. | |
| Jun 22, 2021 at 13:00 | comment | added | Francesco Genovese | @DmitriPavlov no, and the reason is that A_infty categories are not complete/cocomplete. Check §1.5 here arxiv.org/pdf/1811.07830.pdf If you say that there is some "variant" of this, then maybe... but definitely not according to the usual definition | |
| Jun 14, 2021 at 22:49 | comment | added | Dmitri Pavlov | @FrancescoGenovese: The easiest way to construct a model structure on A_∞-categories is to fix a set of objects first. A_∞-categories with a fixed set of objects are algebras over a certain nonsymmetric colored operad, and algebras over nonsymmetric colored operads in chain complexes or simplicial sets always admit a transferred model structure (see Fernando Muro's paper Homotopy theory of nonsymmetric operads, and its errata). | |
| Jun 14, 2021 at 22:01 | comment | added | Fernando Muro | @FrancescoGenovese they do. | |
| Jun 14, 2021 at 15:50 | comment | added | Francesco Genovese | But A_infinity categories don't have a model structure! | |
| Jan 24, 2020 at 5:48 | vote | accept | user142700 | ||
| Jan 24, 2020 at 5:40 | comment | added | Dmitri Pavlov | @user142700: If C=V=Ch, then algebras over the operad Cat^O_W are A_∞-dg-categories respectively dg-categories if O=A_∞ respectively As (the associative operad) with W as a set of objects. | |
| Jan 24, 2020 at 5:38 | comment | added | user142700 | Could you please clarify how Corollary 9.2.1. relates to dg-categories? (I apologize if this is obvious -- I have very little background in this area). | |
| Jan 24, 2020 at 5:28 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |