A-infinity-algebra (changes) in nLab

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Contents

Idea

An A ∞A_\infty-algebra is a monoid internal to a homotopical category such that the associativity law holds not as an equation, but only up to higher coherent homotopy.

Definition

Realizations

In chain complexes

Let here ℰ\mathcal{E} be the category of chain complexes 𝒞𝒽 •\mathcal{Ch}_\bullet. Notice that often in the literature this choice of ℰ\mathcal{E} is regarded as default and silently assumed.

An A ∞A_\infty-algebra in chain complexes is concretely the following data.

A chain A ∞A_\infty-algebra is the structure of a degree 1 coderivation

D:T cV→T cV D : T^c V \to T^c V

on the reduced tensor coalgebra T cV=⊕ n≥1V ⊗nT^c V = \oplus_{n\geq 1} V^{\otimes n} (with the standard noncocommutative coproduct, see differential graded Hopf algebra) over a graded vector space VV such that

D 2=0. D^2 = 0 \,.

Coderivations on free coalgebras are entirely determined by their “value on cogenerators”, which allows one to decompose DD as a sum:

D=D 1+D 2+D 3+⋯ D = D_1 + D_2 + D_3 + \cdots

with each D kD_k specified entirely by its action

D k:V ⊗k→V. D_k : V^{\otimes k} \to V \,.

which is a map of degree 2−k2-k (or can be alternatively understood as a map D k:(V[1]) ⊗k→V[1]D_k : (V[1])^{\otimes k}\to V[1] of degree 11).

Then:

• D 1:V→VD_1 : V\to V is the differential with D 1 2=0D_1^2 = 0;

• D 2:V ⊗2→VD_2 : V^{\otimes 2} \to V is the product in the algebra;

• D 3:V ⊗3→VD_3 : V^{\otimes 3} \to V is the associator which measures the failure of D 2D_2 to be associative;

• D 4:V ⊗4→VD_4 : V^{\otimes 4} \to V is the pentagonator (or so) which measures the failure of D 3D_3 to satisfy the pentagon identity;

• and so on.

One can also allow D 0D_0, in which case one talks about weak A ∞A_\infty-algebras, which are less understood.

There is a resolution of the operad Ass\mathrm{Ass} of associative algebras (as operad on the category of chain complexes) which is called the A ∞A_\infty-operad; the algebras over the A ∞A_\infty-operad are precisely the A ∞A_\infty-algebras.

A morphism of A ∞A_\infty-algebras f:A→Bf : A\to B is a collection {f n} n≥1\lbrace f_n\rbrace_{n\geq 1} of maps

f n:(A[1]) ⊗n→B[1] f_n : (A[1])^{\otimes n}\to B[1]

of degree 00 satisfying

∑ 0≤i+j≤nf i+j+1∘(1 ⊗i⊗D n−i−j⊗1 ⊗j)=∑ i 1+…+i r=nD r∘(f i 1⊗…⊗f i r). \sum_{0\leq i+j\leq n} f_{i+j+1}\circ(1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j}) = \sum_{i_1+\ldots+i_r=n} D_r\circ (f_{i_1}\otimes\ldots \otimes f_{i_r}).

For example, f 1∘D 1=D 1∘f 1f_1\circ D_1 = D_1\circ f_1.

Rectification

More details are at Kadeishvili's theorem.

In Topological space

An A ∞A_\infty-algebra in Top is also called an A-∞ space .

Examples

Every loop space is canonically an A-∞ space. (See there for details.)

Rectification

This is a classical result by (Stasheff 1963, BoardmanVogt). It follows also as a special case of the more general result on rectification in a model structure on algebras over an operad (see there).

In spectra

See ring spectrum and algebra spectrum.

• A ∞A_\infty-algebra, A-∞-category

  • augmented A-∞ algebra

  • curved A-∞ algebra

• An-space

• A-n algebra

• E-∞ algebra

• L-∞ algebra, .

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space ≃\simeq ∞-space	infinite loop space object
				≃\simeq connective spectrum	≃\simeq connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

algebraic deformation quantization

dimension	classical field theory	Lagrangian BV quantum field theory	factorization algebra of observables
general nn	P-n algebra	BD-n algebra?	E-n algebra
n=0n = 0	Poisson 0-algebra	BD-0 algebra? = BD algebra	E-0 algebra? = pointed space
n=1n = 1	P-1 algebra = Poisson algebra	BD-1 algebra?	E-1 algebra? = A-∞ algebra

References

A survey of A ∞A_\infty-algebras in chain complexes is in

• Bernhard Keller, A brief introduction to A ∞A_\infty-algebras (pdf)

Classical articles on A ∞A_\infty-algebra in topological spaces are

• Jim Stasheff, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108 2 (1963) 275-292 [[doi:10.2307/1993608](https://doi.org/10.2307/1993608)]

• Jim Stasheff, Homotopy associativity of H-spaces II 108 2 (1963) 293-312 [[doi:10.2307/1993609](https://doi.org/10.2307/1993609), doi:10.1090/S0002-9947-1963-0158400-5]

• Michael Boardman and Rainer Vogt, Homotopy invariant algebraic structures on topological spaces , Lect. Notes Math. 347 (1973).

A brief survey is in section 1.8 of

• Martin Markl, Steve Shnider, James D. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.

The 1986 thesis of Alain Prouté explores the possibility of obtaining analogues of minimal models for A ∞A_\infty algebras. It was published in TAC much later.

• Alain Prouté, Algèbres différentielles fortement homotopiquement associatives (A ∞A_\infty-algèbres), thesis, available as Reprints in Theory and Applications of Categories, No. 21, 2011, pp. 1–99