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In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the $dd^c$-lemma.
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We study (quasi-)twilled pre-Lie algebras and the associated $L_infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ($L_infty$-algebras). Furthermore, we show that $mathcal{O}$-operators and twisted $mathcal{O}$-operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ($L_infty$-algebras) and the twisting theory of (quasi-)twilled pre-Lie algebras. In particular, we give a construction of quasi-pre-Lie bialgebras using symplectic Lie algebras, which is parallel to that a Cartan $3$-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.
We show that the celebrated operad of pre-Lie algebras is very rigid: it has no non-obvious degrees of freedom from either of the three points of view: deformations of maps to and from the three graces of operad theory, homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005, including a Lie-theoretic version of the Deligne conjecture.
In this note we explain that homotopy coherent simplicial nerve has to used intead of the standard definition in the authors papers on formal deformation theory. A convenient version of the notion of fibered category is presented which is useful once one works with simplicial categories.
We show that Mandells inverse $K$-theory functor is a categorically-enriched multifunctor. In particular, it preserves algebraic structures parametrized by operads. As applications, we describe how ring categories, bipermutative categories, braided ring categories, and $E_n$-monoidal categories arise as the images of inverse $K$-theory.
We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Gamma$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
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