nLab
A-infinity-category

Idea

An $A_{\mathrm{\infty }}$-category is a kind of category in which the associativity condition on the composition of morphisms is relaxed “up to higher coherent homotopy”.

The “A” is for Associative and the ”${}_{\mathrm{\infty }}$” indicates that associativity is relaxed up to higher homotopies without bound on the degree of the homotopies.

In the most widespread use of the word $A_{\mathrm{\infty }}$-categories are linear categories in that they have hom-objects that are chain complexes. These are really models/presentations for stable (∞,1)-categories.

If the composition in the linear $A_{\mathrm{\infty }}$-category does happen to be strictly associative it becomes the same as a dg-category. In fact, every linear $A_{\mathrm{\infty }}$-category is $A_{\mathrm{\infty }}$-equivalent to a dg-category. In this way, we have that $A_{\mathrm{\infty }}$-categories related to dg-categories as models for stable (∞,1)-categories in roughly the same way as quasi-categories relate to simplicially enriched categories as models for (∞,1)-categories: the former is the general incarnation, while the latter is a semi-strictified version.

Ordinary linear $A_{\mathrm{\infty }}$-categories

In what is strictly speaking a restrictive sense – which is however widely and conventionally understood in homological algebra as the standard notion of $A_{\mathrm{\infty }}$-category (see references below) – the hom-spaces of an $A_{\mathrm{\infty }}$-category are taken to be linear spaces, i.e. modules over some ring or field, and in fact chain complexes of such modules.

Therefore an $A_{\mathrm{\infty }}$-category in this standard sense of homological algebra is a category which is in some way homotopically enriched over a category of chain complexes $\mathrm{Ch}$. Since a category which is enriched in the ordinary sense of enriched category theory is a dg-category, there is a close relation between $A_{\mathrm{\infty }}$-categories and dg-categories.

$A_{\mathrm{\infty }}$-categories in this linear sense are a horizontal categorification of the notion of A-infinity-algebra. As such they are to A-infinity-algebras as Lie infinity-algebroids are to L-infinity-algebras. For this point of view see the Kontsevich–Soibelman reference below.

Definition

A category $C$ such that

1. for all $X,Y$ in $\mathrm{Ob}(C)$ the Hom-sets ${\mathrm{Hom}}_C(X,Y)$ are finite dimensional chain complexes of $Z$-graded modules

2. for all objects $X_1,...,X_n$ in $\mathrm{Ob}(C)$ there is a family of linear composition maps (the higher compositions) $m_n:{\mathrm{Hom}}_C(X_0,X_1)\otimes {\mathrm{Hom}}_C(X_1,X_2)\otimes \cdots \otimes {\mathrm{Hom}}_C(X_{n-1},X_n)\to {\mathrm{Hom}}_C(X_0,X_n)$ of degree $n-2$ (homological grading convention is used) for $n\ge 1$

3. $m_1$ is the differential on the chain complex ${\mathrm{Hom}}_C(X,Y)$

4. $m_n$ satisfy the quadratic $A_{\mathrm{\infty }}$-associativity equation for all $n\ge 0$.

$m_1$ and $m_2$ will be chain maps but the compositions $m_i$ of higher order are not chain maps, nevertheless they are Massey product?s.

The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of $A_{\mathrm{\infty }}$-categories and $A_{\mathrm{\infty }}$-functors. Many features of $A_{\mathrm{\infty }}$-categories and $A_{\mathrm{\infty }}$-functors come from the fact that they form a symmetric closed multicategory, which is revealed in the language of comonads.

From a higher dimensional perspective $A_{\mathrm{\infty }}$-categories are weak $\omega$-categories with all morphisms invertible. $A_{\mathrm{\infty }}$-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.

Examples and remarks

• Every dg-category may be regarded as a special case when there are no higher maps (trivial homotopies) of an $A_{\mathrm{\infty }}$-category.

• Every $A_{\mathrm{\infty }}$-category is $A_{\mathrm{\infty }}$-equivalent to a dg-category.

  • This is a corollary of the $A_{\mathrm{\infty }}$-categorical Yoneda lemma.

  • beware that this statement does not imply that the notion of $A_{\mathrm{\infty }}$-categories is obsolete (see section 1.8 in Bespalov et al.): in practice it is often easier to work with a given naturally arising $A_{\mathrm{\infty }}$-category than constructing its equivalent dg-category

    • for instance when dealing with a Fukaya? $A_{\mathrm{\infty }}$-category;

    • or when dealing with various constructions on dg-categories, for instance certain quotients,

  that naturally yield directly $A_{\mathrm{\infty }}$-categories instead of dg-categories.

• The path space of a topological space $X$

• The Fukaya category $\mathrm{Fuk}(X)$ of a topological space $X$ – a Calabi-Yau A-∞ category

• $A_{\mathrm{\infty }}$-algebras as $A_{\mathrm{\infty }}$-categories with one object.

• The loop space $\Omega X$ of a topological space $X$

More general $A_{\mathrm{\infty }}$-categories

In the widest sense, $A_\infty$-category" may be used as a term for a category in which the [[composition]] operation constitutes an algebra over an [[operad]] which resolves in some sense the associative operad $Ass$.

One should be aware, though, that this use of the term is not understood by default in the large body of literature concerned with the above linear notion.

A less general but non-linear definition is fairly straight forward in any category in which there is a notion of homotopy with the usual properties.

Definition

An $A_{\mathrm{\infty }}$-category is a category over the $A_{\mathrm{\infty }}$-operad: e.g. the free resolution in the context of dg-operads of the linear associative operad.

Examples

• Trimble n-categories;

• also the classical notion of bicategory can be interpreted as an $A_{\mathrm{\infty }}$-category in Cat for a suitable Cat-operad.

References

For $A_{\mathrm{\infty }}$-categories in the sense of homological algebra

For a short and precise introduction see

• B. Keller, Introduction to $A_{\mathrm{\infty }}$-algebras and modules (dvi, ps) and Addendum (ps), Homology, Homotopy and Applications 3 (2001), 1-35;

• B. Keller, $A_{\mathrm{\infty }}$ algebras, modules and functor categories, (pdf, ps).

and for a Fukaya category-oriented introduction see chapter 1 in

• P. Seidel, Fukaya category and Picard-Lefschetz theory, draft version

A very detailed treatment of $A_{\mathrm{\infty }}$-categories is a recent book

• Yu. Bespalov, V. Lyubashenko, O. Manzyuk, Pretriangulated $A_{\mathrm{\infty }}$-categories, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 (ps.gz)

  • notice: the ps.gz file has different page numbers than the printed version, but the numbering of sections and formulae is final. Errata to published version are here.

The relation of $A_{\mathrm{\infty }}$-categories to differential graded algebras is emphasized in the introduction of

• Maxim Kontsevich, Yan Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I (arXiv)

which mostly discusses just A-infinity-algebras, but points out a generalizations to $A_{\mathrm{\infty }}$-categories, see the overview on p. 3

Essentially the authors say that an $A_{\mathrm{\infty }}$-category should be a non(-graded-)commutative NQ-supermanifold. Recall that a graded-commutative NQ-supermanifold is a L-infinity-algebroid. From this perspective one would want to think of a non-graded commutative NQ-supermanifold as an $A_{\mathrm{\infty }}$-algebroid.

For $A_{\mathrm{\infty }}$-categories in the wider sense

If one understands $A_{\mathrm{\infty }}$-category as “operadically defined higher category”, then relevant references would include:

• Eugenia Cheng, Comparing operadic definitions of $n$-category (arXiv)