Gauge equivalence for complete $L_{\infty}$-algebras

Guan, Ai

doi:10.48550/arxiv.1807.11932

Cited by 3 publications

(2 citation statements)

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“…A * be induced by dualising the differential and multiplication on A respectively. For a pseudocompact vector space V , consider the semi-completed tensor algebra T ′ (V ) = n≥1 V ⊗n , which has a topology that is neither pseudocompact nor discrete, and has the property Hom Alg (T ′ (V ), B) ∼ = Hom(V, B) for any pseudocompact algebra B (see [Gua,Lemma 4.5]). Then by Proposition 2.2(1), the identity on q…”

Section: Extended Bar Constructionmentioning

confidence: 99%

Koszul duality for compactly generated derived categories of second kind

Guan,

Lazarev

2019

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For any dg algebra A we construct a closed model category structure on dg A-modules such that the corresponding homotopy category is compactly generated by dg A-modules that are finitely generated and free over A (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of A. This generalises and complements certain aspects of dg Koszul duality for associative algebras.

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Section: Extended Bar Constructionmentioning

confidence: 99%

Koszul duality for compactly generated derived categories of second kind

Guan,

Lazarev

2019

Preprint

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“…It is easy to see that Schlessinger-Stasheff theorem 4.5 remains valid in this context. Moreover, Theorem 4.5 has a natural interpretation in terms of model categories: it says, roughly speaking, that the notions of left and right homotopy for nilpotent dg Lie algebras agree (see [10] for a precise statement and its generalizations). 4.2.…”

Section: Elements and MC Moduli Setsmentioning

confidence: 99%

Review of deformation theory II: a homotopical approach

Guan,

Lazarev,

Sheng

et al. 2019

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We give a general treatment of deformation theory from the point of view of homotopical algebra following Hinich, Manetti and Pridham. In particular, we show that any deformation functor in characteristic zero is controlled by a certain differential graded Lie algebra defined up to homotopy, and also formulate a noncommutative analogue of this result valid in any characteristic. Contents1. Introduction 2. Closed model categories 3. Brown representability theorem for compactly generated model categories 4. MC elements and MC moduli sets 4.1. MC moduli in dg Lie algebras 4.2. MC moduli in dg algebras 5. Koszul duality 5.1. DG coalgebras and pseudocompact dg algebras 5.2. Quillen equivalence between DGLA and pcCDGA op loc 5.3. Quillen equivalence between DGA/k and pcDGA op loc 5.4. Relationship between two types of Koszul duality 6. Main theorems 6.1. MC elements and the deformation functor based on a dg Lie algebra 6.2. Finding a dg Lie algebra associated with a deformation functor 6.3. Associative deformation theory 6.4. Finding a dg algebra associated with a deformation functor 6.5. Comparing commutative and associative deformations References

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