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Moduli stacks of algebraic structures and deformation theory Yalin, Sinan in Journal of Noncommutative Geometry (2016) Detailed reference viewed: 63 (0 UL)Moduli stacks of algebraic structures and deformation theory Yalin, Sinan E-print/Working paper (2016) Detailed reference viewed: 55 (0 UL)Maurer-Cartan spaces of filtered L-infinity algebras Yalin, Sinan in Journal of Homotopy & Related Structures (2015) We study several homotopical and geometric properties of Maurer- Cartan spaces for L-infinity algebras which are not nilpotent, but only filtered in a suitable way. Such algebras play a key role ... [more ▼] We study several homotopical and geometric properties of Maurer- Cartan spaces for L-infinity algebras which are not nilpotent, but only filtered in a suitable way. Such algebras play a key role especially in the deformation theory of algebraic structures. In particular, we prove that the Maurer-Cartan simplicial set preserves fibrations and quasi-isomorphisms. Then we present an algebraic geometry viewpoint on Maurer-Cartan moduli sets, and we compute the tangent complex of the associated algebraic stack. [less ▲] Detailed reference viewed: 44 (4 UL)Realization spaces of algebraic structures on chains Yalin, Sinan E-print/Working paper (2015) Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology ... [more ▼] Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology level. Our algebraic structures are parametrised by props and thus include various kinds of bialgebras. We give a general formula to compute subsets of equivalences classes of realizations as quotients of automorphism groups, and determine the higher homotopy groups via the cohomology of deformation complexes. As a motivating example, we compute subsets of equivalences classes of realizations of Poincar\'e duality for several examples of manifolds. [less ▲] Detailed reference viewed: 29 (0 UL)Function spaces and classifying spaces of algebras over a prop Yalin, Sinan E-print/Working paper (2015) The goal of this paper is to prove that the classifying spaces of categories of algebras governed by a prop can be determined by using function spaces on the category of props. We first consider a ... [more ▼] The goal of this paper is to prove that the classifying spaces of categories of algebras governed by a prop can be determined by using function spaces on the category of props. We first consider a function space of props to define the moduli space of algebra structures over this prop on an object of the base category. Then we mainly prove that this moduli space is the homotopy fiber of a forgetful map of classifying spaces, generalizing to the prop setting a theorem of Rezk. The crux of our proof lies in the construction of certain universal diagrams in categories of algebras over a prop. We introduce a general method to carry out such constructions in a functorial way. [less ▲] Detailed reference viewed: 66 (0 UL)Moduli stacks of algebraic structures and deformation theory Yalin, Sinan in Journal of Noncommutative Geometry (2015) Detailed reference viewed: 75 (1 UL)Classifying spaces of algebras over a prop Yalin, Sinan in Algebraic and Geometric Topology (2014), 14(5), 2561--2593 We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy ... [more ▼] We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop leads us to introduce new methods to overcome this difficulty. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes. [less ▲] Detailed reference viewed: 74 (3 UL)The homotopy theory of bialgebras over pairs of operads Yalin, Sinan in Journal of Pure and Applied Algebra (2014), 218(6), 973-991 We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic ... [more ▼] We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second step we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes. [less ▲] Detailed reference viewed: 63 (0 UL)Simplicial localization of homotopy algebras over a prop Yalin, Sinan in Mathematical Proceedings of the Cambridge Philosophical Society (2014), 157(3), 457468 We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras ... [more ▼] We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories. [less ▲] Detailed reference viewed: 20 (0 UL) |
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