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'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
over a commutative ring ''R'', with an operad replacing ''R''.
Definitions
Given an operad ''O'' (say, a
symmetric sequence 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, 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 \cdo ...
*
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 hom ...
Notes
References
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External links
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*http://ncatlab.org/nlab/show/algebra+over+an+operad
Abstract algebra
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