nLab A-infinity-category

Idea

An $A_\infty$-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 “${}_\infty$” 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 $A_\infty$-categories are linear categories in that they have hom-objects that are chain complexes. These are really models/presentations for stable (∞,1)-categories.

If higher coherences in a linear $A_\infty$-category happen to be equal to identity morphisms (which is encoded by the vanishing of the maps $m_n$ for $n\ge3$, defined below), it becomes the same as a dg-category. In fact, every linear $A_\infty$-category is $A_\infty$-equivalent to a dg-category. In this way, we have that $A_\infty$-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 $A_\infty$-categories

In what is strictly speaking a restrictive sense – which is however widely and conventionally understood in homological algebra as the standard notion of $A_\infty$-category (see references below) – the hom-spaces of an $A_\infty$-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 $A_\infty$-category in this standard sense of homological algebra is a category which is in some way homotopically enriched over a category of chain complexes $Ch$. Since a category which is enriched in the ordinary sense of enriched category theory is a dg-category, there is a close relation between $A_\infty$-categories and dg-categories.

$A_\infty$-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 Konsevich-Soibelman 08.

Definition

A category $C$ such that

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

2. for all objects $X_1,...,X_n$ in $Ob(C)$ there is a family of linear composition maps (the higher compositions) $m_n : Hom_C(X_0,X_1) \otimes Hom_C(X_1,X_2) \otimes \cdots \otimes Hom_C(X_{n-1},X_n) \to Hom_C(X_0,X_n)$ of degree $n-2$ (homological grading convention is used) for $n\geq1$

3. $m_1$ is the differential on the chain complex $Hom_C(X,Y)$

4. $m_n$ satisfy the quadratic $A_\infty$-associativity equation for all $n\geq0$.

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

The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of $A_\infty$-categories and $A_\infty$-functors. Many features of $A_\infty$-categories and $A_\infty$-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 $A_\infty$-categories are weak $\omega$-categories with all morphisms invertible. $A_\infty$-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 $A_\infty$-category.

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

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

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

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

    • or when dealing with various constructions on dg-categories, for instance certain quotients,that naturally yield directly $A_\infty$-categories instead of dg-categories.

• The path space of a topological space $X$

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

• $A_\infty$-algebras are the $A_\infty$-categories with one object.

  • For example, the delooping $\mathbf{B}\Omega{X}$ of loop space $\Omega{X}$ of a topological space $X$

More general $A_\infty$-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 $A_\infty$-category is a category over the $A_\infty$-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 $A_\infty$-category in Cat for a suitable Cat-operad.

Related concepts

• A-∞ space, associahedron

• A-∞ algebra

• stable (∞,1)-category

References

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

For a short and precise introduction see

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

• B. Keller, $A_\infty$ 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

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

• Yu. Bespalov, Volodymyr Lyubashenko, O. Manzyuk, Pretriangulated $A_\infty$-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.

• Oleksandr Manzyk, A-infinity-bimodules and Serre A-infinity-functors, dissertation pdf, djvu; Serre $A_\infty$ functors, talk at Categories in geometry and math. physics, Split 2007, slides, pdf, work with Volodymyr Lyubashenko

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

• Maxim Kontsevich, Yan Soibelman, Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry. I, in Anton Kapustin, Maximilian Kreuzer, Karl-Georg Schlesinger (eds.) Homological mirror symmetry – New developments and perspectives, Springer 2008 (arXiv:math/0606241, doi:10.1007/978-3-540-68030-7)

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

Essentially the authors say that an $A_\infty$-category should be a non(-graded-)commutative dg-manifold/L-infinity-algebroid.

More category theory and homotopy theory of $A_\infty$-categories is discussed in

• Kenji Lefèvre-Hasegawa, Sur les A-infini catégories (arXiv:math/0310337)

• Bruno Valette?, Homotopy theory of homotopy algebras, Annales de l’Institut Fourier 70:2 (2020) 683–738 doi Zbl:1335.18001 arXiv:1411.5533

See also

• Yong-Geun Oh, Hiro Lee Tanaka, $A_\infty$-categories, their $\infty$-categories, and their localizations (arXiv:2003.05806)

For $A_\infty$-categories in the wider sense

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

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

With operads modeled by dendroidal sets, n-categories for low $n$ viewed as objects with an $A-\infty$-composition operation are discussed in section 5 of

• Andor Lucacs, Cyclic Operads, Dendroidal Structures, Higher Categories (pdf)

and

• Andor Lucacs, Dendroidal weak 2-categories (arXiv:1304.4278)

See also the references at model structure on algebras over an operad.