Homotopy associative algebra

In mathematics, an algebra such as formula_1 has multiplication formula_2 whose associativity is well-defined on the nose. This means for any real numbers formula_3 we have
But, there are algebras formula_5 which are not necessarily associative, meaning if formula_6 then
in general. There is a notion of algebras, called formula_8-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. This means although we get something which looks like the second equation, the one of inequality, we actually get equality after "compressing" the information in the algebra.
The study of formula_8-algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algebras through a differential graded algebra with a multiplication operation and a series of higher homotopies giving the failure for the multiplication to be associative. Loosely, an formula_8-algebra formula_11 is a formula_12-graded vector space over a field formula_13 with a series of operations formula_14 on the formula_15-th tensor powers of formula_16. The formula_17 corresponds to a chain complex differential, formula_18 is the multiplication map, and the higher formula_14 are a measure of the failure of associativity of the formula_18. When looking at the underlying cohomology algebra formula_21, the map formula_18 should be an associative map. Then, these higher maps formula_23 should be interpreted as higher homotopies, where formula_24 is the failure of formula_18 to be associative, formula_26 is the failure for formula_24 to be higher associative, and so forth. Their structure was originally discovered by Jim Stasheff while studying A∞-spaces, but this was interpreted as a purely algebraic structure later on. These are spaces equipped with maps that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so forth.
They are ubiquitous in homological mirror symmetry because of their necessity in defining the structure of the Fukaya category of D-branes on a Calabi–Yau manifold who have only a homotopy associative structure.
Definition.
Definition.
For a fixed field formula_13 an formula_8-algebra is a formula_12-graded vector space
such that for formula_32 there exist degree formula_33, formula_13-linear maps
which satisfy a coherence condition:
where formula_37.
Understanding the coherence conditions.
The coherence conditions are easy to write down for low degreespgs 583–584.
d=1.
For formula_38 this is the condition that
since formula_40 giving formula_41 and formula_42. These two inequalities force formula_43 in the coherence condition, hence the only input of it is from formula_44. Therefore formula_45 represents a differential.
d=2.
Unpacking the coherence condition for formula_46 gives the degree formula_47 map formula_48. In the sum there are the inequalities
of indices giving formula_50 equal to formula_51. Unpacking the coherence sum gives the relation
which when rewritten with
as the differential and multiplication, it is
which is the Leibniz rule for differential graded algebras.
d=3.
In this degree the associativity structure comes to light. Note if formula_56 then there is a differential graded algebra structure, which becomes transparent after expanding out the coherence condition and multiplying by an appropriate factor of formula_57, the coherence condition reads something like
Notice that the left hand side of the equation is the failure for formula_18 to be an associative algebra on the nose. One of the inputs for the first three formula_24 maps are coboundaries since formula_17 is the differential, so on the cohomology algebra formula_62 these elements would all vanish since formula_63. This includes the final term formula_64 since it is also a coboundary, giving a zero element in the cohomology algebra. From these relations we can interpret the formula_24 map as a failure for the associativity of formula_18, meaning it is associative only up to homotopy.
d=4 and higher order terms.
Moreover, the higher order terms, for formula_67, the coherent conditions give many different terms combining a string of consecutive formula_68 into some formula_69 and inserting that term into an formula_70 along with the rest of the formula_71's in the elements formula_72. When combining the formula_17 terms, there is a part of the coherence condition which reads similarly to the right hand side of formula_74, namely, there are terms
In degree formula_76 the other terms can be written out as
showing how elements in the image of formula_24 and formula_18 interact. This means the homotopy of elements, including one that's in the image of formula_18 minus the multiplication of elements where one is a homotopy input, differ by a boundary. For higher order formula_81, these middle terms can be seen how the middle maps formula_82 behave with respect to terms coming from the image of another higher homotopy map.
Diagrammatic interpretation of axioms.
There is a nice diagrammatic formalism of algebras which is described in explaining how to visually think about this higher homotopies. This intuition is encapsulated with the discussion above algebraically, but it is useful to visualize it as well.
Examples.
Associative algebras.
Every associative algebra formula_83 has an formula_8-infinity structure by defining formula_85 and formula_86 for formula_87. Hence formula_8-algebras generalize associative algebras.
Differential graded algebras.
Every differential graded algebra formula_89 has a canonical structure as an formula_8-algebra where formula_91 and formula_18 is the multiplication map. All other higher maps formula_14 are equal to formula_94. Using the structure theorem for minimal models, there is a canonical formula_8-structure on the graded cohomology algebra formula_96 which preserves the quasi-isomorphism structure of the original differential graded algebra. One common example of such dga's comes from the Koszul algebra arising from a regular sequence. This is an important result because it helps pave the way for the equivalence of homotopy categoriesformula_97of differential graded algebras and formula_8-algebras.
Cochain algebras of H-spaces.
One of the motivating examples of formula_8-algebras comes from the study of H-spaces. Whenever a topological space formula_100 is an H-space, its associated singular chain complex formula_101 has a canonical formula_8-algebra structure from its structure as an H-space.
Example with infinitely many non-trivial mi.
Consider the graded algebra formula_103 over a field formula_13 of characteristic formula_94 where formula_106 is spanned by the degree formula_94 vectors formula_108 and formula_109 is spanned by the degree formula_110 vector formula_111. Even in this simple example there is a non-trivial formula_8-structure which gives differentials in all possible degrees. This is partially due to the fact there is a degree formula_110 vector, giving a degree formula_13 vector space of rank formula_110 in formula_116. Define the differential formula_17 by
and for formula_119
where formula_121 on any map not listed above and formula_122. In degree formula_123, so for the multiplication map, we have
formula_124
And in formula_74 the above relations give
When relating these equations to the failure for associativity, there exist non-zero terms. For example, the coherence conditions for formula_127 will give a non-trivial example where associativity doesn't hold on the nose. Note that in the cohomology algebra formula_128 we have only the degree formula_94 terms formula_108 since formula_111 is killed by the differential formula_17.
Properties.
Transfer of A structure.
One of the key properties of formula_8-algebras is their structure can be transferred to other algebraic objects given the correct hypotheses. An early rendition of this property was the following: Given an formula_8-algebra formula_16 and a homotopy equivalence of complexes
then there is an formula_8-algebra structure on formula_138 inherited from formula_16 and formula_140 can be extended to a morphism of formula_8-algebras. There are multiple theorems of this flavor with different hypotheses on formula_138 and formula_140, some of which have stronger results, such as uniqueness up to homotopy for the structure on formula_138 and strictness on the map formula_140.
Structure.
Minimal models and Kadeishvili's theorem.
One of the important structure theorems for formula_8-algebras is the existence and uniqueness of minimal models – which are defined as formula_8-algebras where the differential map formula_148 is zero. Taking the cohomology algebra formula_96 of an formula_8-algebra formula_16 from the differential formula_17, so as a graded algebra,
with multiplication map formula_154. It turns out this graded algebra can then canonically be equipped with an formula_8-structure,
which is unique up-to quasi-isomorphisms of formula_8-algebras. In fact, the statement is even stronger: there is a canonical formula_8-morphism
which lifts the identity map of formula_16. Note these higher products are given by the Massey product.
Motivation.
This theorem is very important for the study of differential graded algebras because they were originally introduced to study the homotopy theory of rings. Since the cohomology operation kills the homotopy information, and not every differential graded algebra is quasi-isomorphic to its cohomology algebra, information is lost by taking this operation. But, the minimal models let you recover the quasi-isomorphism class while still forgetting the differential. There is an analogous result for A∞-categories by Maxim Kontsevich and Yan Soibelman, giving an A∞-category structure on the cohomology category formula_161 of the dg-category consisting of cochain complexes of coherent sheaves on a non-singular variety formula_100 over a field formula_13 of characteristic formula_94 and morphisms given by the total complex of the Cech bi-complex of the differential graded sheaf formula_165pg 586-593. In this was, the degree formula_13 morphisms in the category formula_161 are given by formula_168.
Applications.
There are several applications of this theorem. In particular, given a dg-algebra, such as the de Rham algebra formula_169, or the Hochschild cohomology algebra, they can be equipped with an formula_8-structure.
Massey structure from DGA's.
Given a differential graded algebra formula_89 its minimal model as an formula_8-algebra formula_156 is constructed using the Massey products. That is,
It turns out that any formula_8-algebra structure on formula_96 is closely related to this construction. Given another formula_8-structure on formula_96 with maps formula_179, there is the relation
where
Hence all such formula_8-enrichments on the cohomology algebra are related to one another.
Graded algebras from its ext algebra.
Another structure theorem is the reconstruction of an algebra from its ext algebra. Given a connected graded algebra
it is canonically an associative algebra. There is an associated algebra, called its Ext algebra, defined as
where multiplication is given by the Yoneda product. Then, there is an formula_8-quasi-isomorphism between formula_186 and formula_184. This identification is important because it gives a way to show that all derived categories are derived affine, meaning they are isomorphic the derived category of some algebra.