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Algebra Over An Operad
In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring ''R'', with an operad replacing ''R''. Definitions Given an operad ''O'' (say, a symmetric sequence in a symmetric monoidal ∞-category ''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 f: O \to O' 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 *Homotopy Lie algebra In mathematics, in particular abstract algebra and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Interest in operads was consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 category with a single object to ''C'' that maps the single object to ''X''. Note this functor then induces a group homomorphism G \to \operatorname(X); cf. Automorphism group#In category theory. of the symmetric group \mathbb_n. The category of combinatorial species is equivalent to the category of finite \mathbb-sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.) S-module By ''\mathbb-module'', we mean an \mathbb-object in the category \mathsf of finite-dimensional vector spaces over a field ''k'' of characteristic zero (the symmetric groups act from the right by convention). Then each \mathbb-module determines a Schur functor on \mathsf. This definition of \mathbb-module shares i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 sense, naturally isomorphic to B\otimes A for all objects A and B of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field ''k,'' using the ordinary tensor product of vector spaces. Definition A symmetric monoidal category is a monoidal category (''C'', ⊗, ''I'') such that, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism s_: A \otimes B \to B \otimes A that is natural in both ''A'' and ''B'' and such that the following diagrams commute: *The unit coherence: *: *The associativity coherence: *: *The inverse law: *: In the diagrams above, ''a'', ''l'' , ''r'' are the associativity isomorphism, the left unit isomorphism, and the right un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Interest in operads was consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 \cdots \otimes A(U_m) \to A(V) :for any disjoint open disks U_j contained in some open disk ''V'' subject to the requirements that the multiplication maps are compatible with composition, and that \mu is an equivalence if m=1. An equivalent definition is that ''A'' is an algebra in ''C'' over the little ''n''-disks operad. Examples * An \mathcal_n-algebra in vector spaces over a field is a unital associative algebra if ''n'' = 1, and a unital commutative associative algebra if ''n'' ≥ 2. * An \mathcal_n-algebra in categories is a monoidal category if ''n'' = 1, a braided monoidal category if ''n'' = 2, and a symmetric monoidal category if ''n'' ≥ 3. * If Λ is a commutative ring, then X \mapsto C_*(\Omega^n X; \Lambda) defines an \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 homotopy. Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in deformation theory because deformation functors are classified by quasi-isomorphism classes of L_\infty-algebras. This was later extended to all characteristics by Jonathan Pridham. Homotopy Lie algebras have applications within mathematics and mathematical physics; they are linked, for instance, to the Batalin–Vilkovisky formalism much like differential graded Lie algebras are. Definition There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
