The first obstructions to enhancing a triangulated category
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Mathematische Zeitschrift
The first obstructions to enhancing a triangulated category Fernando Muro1 Received: 13 September 2018 / Accepted: 6 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an example of triangulated category without models. We also obtain cohomological characterizations of pre-triangulated DG, A-infinity, and spectral categories. Mathematics Subject Classification 18E30 · 16E40 · 18G50 · 18G60
Contents 1 Introduction . . . . . . . . . . . . . . . . . . 2 Hochschild cohomology of categories . . . . . 3 Heller’s classification of triangulated structures 4 Toda brackets . . . . . . . . . . . . . . . . . . 5 A local-to-global spectral sequence . . . . . . 6 The octahedral axiom . . . . . . . . . . . . . 7 The topological case . . . . . . . . . . . . . . 8 Example of non-vanishing obstructions . . . . References . . . . . . . . . . . . . . . . . . . . .
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1 Introduction Heller [15] noted that a triangulated structure on an essentially small additive category T with suspension functor : T → T induces a stable Toda bracket partial composition-like operation, sending morphisms
The author was partially supported by the Spanish Ministry of Economy under the Grant MTM2016-76453-C2-1-P (AEI/FEDER, UE).
B 1
Fernando Muro [email protected] http://personal.us.es/fmuro Facultad de Matemáticas, Departamento de Álgebra, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Seville, Spain
123
F. Muro f
g
h
X −→ Y −→ Z −→ T
(1.1)
with g f = 0 and hg = 0 to a coset h, g, f ⊂ T (X , −1 T ), satisfying certain properties. Exact triangles f
i
q
X −→ Y −→ C f −→ X are characterized by the fact that the Toda bracket contains the identity map id X ∈ q, i, f ⊂ T (X , X ). The graded category T associated with the pair (T , ) is given by Tn (X , Y ) = T (X , n Y ), n ∈ Z.
An algebraic enhancement of T in the sense of Bondal and Kapranov [5] is a DG-category C with H ∗ (C ) = T such that the previous Toda brackets coincide with the standard Massey products in the cohomology of C . Assume we are working over a field k. By Kadeishvili’s theorem [18,22], a Bondal– Kapranov enhancement is essentially the same as a minimal A-infinity category structure on T . The first pos
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