-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.
-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.
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.
(In both coherence conditions, the signs in the sums can be bypassed by shifting the grading by one.)
The coherence conditions are easy to write down for low degrees.
In this degree the coherence condition reads Notice that the left hand side of the equation is the failure of the multiplication
In higher degrees the coherency conditions give many different terms.
We can arrange the right hand side to be a chain homotopy given by
: while the terms on the left hand side indicate the failure of lower
terms to satisfy a kind of generalized associativity.
There is a nice diagrammatic formalism of algebras which is described in Algebra+Homotopy=Operad[5] 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.
-algebra requires an infinite sequence of higher multiplications, one might hope that there is a way to repackage the definition in terms of a single structure with finitely many operations.
with the standard tensor product symbol and the external tensor product used in defining a coproduct with the vertical stroke for clarity.
-algebras is equivalently a quasiisomorphism of differential graded coalgebras, and a homotopy between
Using the structure theorem for minimal models, there is a canonical
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 categories
is an H-space, its associated singular chain complex
will give a non-trivial example where associativity doesn't hold on the nose.
-algebras is their structure can be transferred to other algebraic objects given the correct hypotheses.
, some of which have stronger results, such as uniqueness up to homotopy for the structure on
-algebras is the existence and uniqueness of minimal models – which are defined as
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
of the dg-category consisting of cochain complexes of coherent sheaves on a non-singular variety
and morphisms given by the total complex of the Cech bi-complex of the differential graded sheaf