## Operads, homotopy algebra, and iterated integrals for double loop spaces (1995)

Venue: | 15 T. KASHIWABARA – ON THE HOMOTOPY TYPE OF CONFIGURATION COMPLEXES, CONTEMP. MATH. 146 |

Citations: | 26 - 0 self |

### BibTeX

@INPROCEEDINGS{Getzler95operads,homotopy,

author = {E. Getzler and J. D. S. Jones},

title = {Operads, homotopy algebra, and iterated integrals for double loop spaces },

booktitle = {15 T. KASHIWABARA – ON THE HOMOTOPY TYPE OF CONFIGURATION COMPLEXES, CONTEMP. MATH. 146},

year = {1995},

pages = {159--170},

publisher = {}

}

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### Abstract

Chen's theory of iterated integrals provides a remarkable model for the differential forms on the based loop space M of a differentiable manifold M (Chen [10]; see also Hain-Tondeur [23] and Getzler-Jones-Petrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of differentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding two-dimensional topological field theories. Our approach is to use the formalism of operads. Operads can be defined in any symmetric monoidal category, although we will mainly be concerned with dg-operads (differential graded operads), that is, operads in the category of chain complexes with monoidal structure defined by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative | we give a precise definition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a non-linear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the k-th tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or a-algebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)