# nLab A-infinity-category

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

### Theorems

#### Higher category theory

higher category theory

# Contents

## 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 the composition in the linear ${A}_{\infty }$-category does happen to be strictly associative 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 $\mathrm{Ch}$. Since a category which is enriched in the ordinary sense of enriched category theory is a dg-category, there is a close relation between ${A}_{\infty }$-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 the Kontsevich–Soibelman reference below.

###### Definition

A category $C$ such that

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

2. for all objects ${X}_{1},...,{X}_{n}$ in $\mathrm{Ob}\left(C\right)$ there is a family of linear composition maps (the higher compositions) ${m}_{n}:{\mathrm{Hom}}_{C}\left({X}_{0},{X}_{1}\right)\otimes {\mathrm{Hom}}_{C}\left({X}_{1},{X}_{2}\right)\otimes \cdots \otimes {\mathrm{Hom}}_{C}\left({X}_{n-1},{X}_{n}\right)\to {\mathrm{Hom}}_{C}\left({X}_{0},{X}_{n}\right)$ of degree $n-2$ (homological grading convention is used) for $n\ge 1$

3. ${m}_{1}$ is the differential on the chain complex ${\mathrm{Hom}}_{C}\left(X,Y\right)$

4. ${m}_{n}$ satisfy the quadratic ${A}_{\infty }$-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 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 $\mathrm{Fuk}\left(X\right)$ 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 $B\Omega X$ 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 $\mathrm{Ass}$.

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

## 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, draft version

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

• Yu. Bespalov, V. 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

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

### 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