In algebra, an operad algebra is an "algebra" over an
operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
. It is a generalization of an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
over a commutative ring ''R'', with an operad replacing ''R''.
Definitions
Given an operad ''O'' (say, a
symmetric sequence In algebraic topology, an \mathbb-object (also called a symmetric sequence) is a sequence \ of objects such that each X(n) comes with an actionAn action of a group ''G'' on an object ''X'' in a category ''C'' is a functor from ''G'' viewed as a cate ...
in a
symmetric monoidal ∞-category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...
''C''), an algebra over an operad, or ''O''-algebra for short, is, roughly, a left module over ''O'' with multiplications parametrized by ''O''.
If ''O'' is a
topological operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...
, then one can say an algebra over an operad is an ''O''-monoid object in ''C''. If ''C'' is symmetric monoidal, this recovers the usual definition.
Let ''C'' be symmetric monoidal ∞-category with monoidal structure distributive over colimits. If
is a map of operads and, moreover, if ''f'' is a homotopy equivalence, then the ∞-category of algebras over ''O'' in ''C'' is equivalent to the ∞-category of algebras over ''O in ''C''.
See also
*
En-ring
In mathematics, an \mathcal_n-algebra in a symmetric monoidal infinity category ''C'' consists of the following data:
*An object A(U) for any open subset ''U'' of Rn homeomorphic to an ''n''-disk.
*A multiplication map:
*:\mu: A(U_1) \otimes \cd ...
*
Homotopy Lie algebra In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L_\infty-algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to ho ...
Notes
References
*
*
External links
*http://ncatlab.org/nlab/show/operad
*http://ncatlab.org/nlab/show/algebra+over+an+operad
Abstract algebra
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