Related papers: $A_\infty$ structures and Massey products
We study the construction of tensor products of representations up to homotopy, which are the A-infinity version of ordinary representations. We provide formulas for the construction of tensor products of representations up to homotopy and…
We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable $L_{\infty}$-algebra morphisms. On the "semi-direct product" we construct a homological vector field that projects to the Lie algebroid. Our main…
We provide a differential structure on arbitrary cleft extensions $B:=A^{\mathrm{co}H}\subseteq A$ for an $H$-comodule algebra $A$. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra…
This master's thesis contains an introduction to $A_\infty$-algebras and homological perturbation theory. We then discuss the formality of compact K\"ahler manifolds and present a direct proof of a homotopy transfer principle of…
We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. Therefore,…
Higher structures - infinity algebras and other objects up to homotopy, categorified algebras, `oidified' concepts, operads, higher categories, higher Lie theory, higher gauge theory... - are currently intensively investigated in…
The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product…
We present a deformation theory associated to the higher Hochschild cohomology $H_{S^2}^*(A,A)$. We also study a $G$-algebra structure associated to this deformation theory.
Transfer systems on finite posets have recently been gaining traction as a key ingredient in equivariant homotopy theory. Additionally, they also naturally occur in the data of a model structure. We give a complete characterization of all…
We develop a homotopy theory of $L_\infty$ algebras based on the Lawrence-Sullivan construction, a complete differential graded Lie algebra which, as we show, satisfies the necessary properties to become the right cylinder in this category.…
Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a…
We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a…
Let $M$ be a closed, oriented, simply connected 6-manifold. After localization away from 2, we give a homotopy decomposition of $\Sigma M$ in terms of spheres, Moore spaces and other recognizable spaces. As applications we calculate…
We show that an A-infinity algebra structure can be transferred to a projective resolution of the complex underlying any A-infinity algebra. Under certain connectedness assumptions, this transferred structure is unique up to homotopy. In…
In this paper we analyze some relationships between the topological complexity of a space $X$ and the category of $C_{\Delta_X},$ the homotopy cofibre of the diagonal map $\Delta_X:X\rightarrow X\times X.$ We establish the equality of the…
Given a commutative algebra $A$ and a quotient $A$-algebra $A/I$, we construct a resolution of $A/I$ as an $A$-module such that it is also a differential graded (dg) algebra with divided powers (PD). This construction makes use of symmetric…
In this review we give a detailed introduction to the theory of (curved) $L_\infty$-algebras and $L_\infty$-morphisms. In particular, we recall the notion of (curved) Maurer-Cartan elements, their equivalence classes and the twisting…
We define a new invariant in the homology of a differential graded algebra. This invariant is the obstruction to defining a fourfold Massey product.
We give a proof of the Homotopy Transfer Theorem following Kadeishvili's original strategy. Although Kadeishvili originally restricted himself to transferring a dg algebra structure to an $A_\infty$-structure on homology, we will see that a…
We show that the tensor product of two cyclic $A_\infty$-algebras is, in general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the…