Homotopy associative algebra¶
There is a notion of algebras, called A_\infty -algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative.
Core Idea¶
Homotopy associative algebra is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: There is a notion of algebras, called A\infty -algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative. In mathematics, an algebra such as (\R,+,\cdot) has multiplication \cdot whose associativity is well-defined on the nose.
Scope of Application¶
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Diagrammatic interpretation of axioms. Given a (graded) vector space V , the reduced tensor coalgebra \overline{T^c}V on V is \bigoplus{n \geq 1} V^{\otimes n} with the (non-cocommutative) coproduct.
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Applications. There are several applications of this theorem.
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DefinitionDefinition. For a fixed field k an A\infty -algebra is a \Z -graded vector space.
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DefinitionDefinition. equipped with morphisms mi \colon A^{\otimes i} \to A of degree 2 - i for each i \geq 1 satisfying a coherence condition: for all n ,.
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DefinitionDefinition. \sum{j+k+l=n,\,j+1+l=i} (-1)^{jk+l} mi(\mathrm{id}^{\otimes j} \otimes mk \otimes \mathrm{id}^{\otimes l}) = 0 .
Clarity¶
A clear use of Homotopy associative algebra names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is There is a notion of algebras, called A\infty -algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say.
Manages Complexity¶
Homotopy associative algebra compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—the right hand side is the differential on A applied to the triple product plus the triple product applied to the differential on A \otimes A \otimes A , and says precisely that associativity holds up to a homotopy given by m3 .—and the practical consequence—since the definition of an.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: There is a notion of algebras, called A\infty -algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative.
- Check operation and conditions.
Knowledge Transfer¶
Within the home domain. Knowledge about Homotopy associative algebra transfers literally when a new case preserves the same carrier type, relation, and recognition test. Given a (graded) vector space V , the reduced tensor coalgebra \overline{T^c}V on V is \bigoplus{n \geq 1} V^{\otimes n} with the (non-cocommutative) coproduct \Delta{\overline{T^c}} given by splitting of tensors, i.e., \Delta(v1 \otimes \cdots \otimes vn) = \sum{1 \leq i , where we write the internal tensor product of \overline{T^c}V with the standard tensor product symbol and the external tensor product used in defining a.
Neighborhood in Abstraction Space¶
Homotopy associative algebra sits in a crowded region of the domain-specific corpus (39th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Category Theory & Homotopical Algebra (18 abstractions)
Nearest neighbors
- Filling radius — 0.88
- Two-Element Boolean Algebra — 0.87
- Linear order — 0.87
- A∞-operad — 0.87
- S-procedure — 0.87
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