Remark 2.1. After the completion of this paper, the paper [L5] was revised to include a thorough treatment of -categorical operads and their algebras. Furthermore, the paper [L7] studies in great detail the specific case of the -operads (including a proof of general versions of the Deligne-Kontsevich conjecture). We refer the reader to these preprints for details on these topics.
2.5. -structures
When considering monoidal structures on vector spaces, we have the option of considering associative or commutative multiplications. When considering monoidal structures on categories, we are faced with three levels of increasing commutativity: “plain” monoidal categories carrying an associative multiplication; braided monoidal categories, in which there is a functorial isomorphism exchanging the order of multiplication and satisfying the braid relations; and symmetric monoidal categories, in which the square of the braiding is the identity.
In homotopy theory, there is an infinite sequence of types of algebraic structures interpolating between associativity and commutativity, modeled on the increasing commutativity of -fold iterated loop spaces. These algebraic structures can be encoded by the little -disk or operad for or , which is an operad in the category of topological spaces. Recall that an operad (in spaces) is a sequence of spaces , which parametrize -fold multiplication operations in the algebraic structures we are encoding, together with actions of the symmetric group permuting the entries and equivariant composition maps. The -th space of the operad parametrizes disjoint collections of small balls in the -ball, with the natural “picture-in-picture” composition maps. An -vector space carries operations labelled by components of the operad, and is an associative algebra for and a commutative algebra for . The notion of -category for a (usual discrete) category is sensitive to the fundamental groupoid of the operad, leading to the three different notions of monoidal category (), braided monoidal category () and symmetric monoidal category (, since for the spaces in the operad are simply connected). However, even on the level of graded vector spaces, with operations labelled by the homology of the operad, we obtain different notions for every , with giving the notion of a Gerstenhaber algebra familiar from the study of Hochschild cohomology.
We have already encountered the notions of algebra object (corresponding to the case , or equivalently ) and commutative algebra object (corresponding to ) in the context of a symmetric monoidal -category, such as dg modules over a ring , spectra or presentable -categories. In [F1], the general theory of algebras over operads in -categories is developed and applied to algebra and geometry in the case. Roughly speaking, in the -categorical context, we consider the operadic operations and compositions in a homotopy coherent fashion (see Section 5.3 for more details). Thus for example, an -category is a homotopy-theoretic analogue of a braided monoidal category.
The notions of -algebras and categories pervade homotopy theory, but have also become prominent in algebra and topological field theory. We briefly mention two such applications.
Deligne’s Hochschild cohomology conjecture asserts that the Hochschild cochain complex of an associative algebra is an -algebra, lifting the Gerstenhaber algebra structure on Hochschild cohomology. Kontsevich’s conjecture generalizes this to assert that the Hochschild cohomology of an -algebra is an -algebra.
The space of states associated to an -sphere by an -dimensional topological field theory has a natural -structure, given by tree-level field theory operations, independent of where the field theory takes its values (vector spaces, chain complexes, categories, etc.)
Original source: arXiv:0805.0157v5