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A-infinity-category

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Idea

An A 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 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 A_\infty-category does happen to be strictly associative it becomes the same as a dg-category. In fact, every linear A A_\infty-category is A A_\infty-equivalent to a dg-category. In this way, we have that A 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 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 A_\infty-category (see references below) – the hom-spaces of an A 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 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 ChCh. 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 A_\infty-categories and dg-categories.

A 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 CC such that

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

  2. for all objects X 1,...,X nX_1,...,X_n in Ob(C)Ob(C) there is a family of linear composition maps (the higher compositions) m n:Hom C(X 0,X 1)Hom C(X 1,X 2)Hom C(X n1,X n)Hom C(X 0,X n)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 n2n-2 (homological grading convention is used) for n1n\geq1

  3. m 1m_1 is the differential on the chain complex Hom C(X,Y)Hom_C(X,Y)

  4. m nm_n satisfy the quadratic A A_\infty-associativity equation for all n0n\geq0.

m 1m_1 and m 2m_2 will be chain maps but the compositions m im_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 A_\infty-categories and A A_\infty-functors. Many features of A A_\infty-categories and A 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 A_\infty-categories are weak ω\omega-categories with all morphisms invertible. A 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 A_\infty-category.

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

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

    • beware that this statement does not imply that the notion of A 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 A_\infty-category than constructing its equivalent dg-category

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

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

  • The path space of a topological space XX

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

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

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

More general A A_\infty-categories

In the widest sense, A 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 AssAss.

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 A_\infty-category is a category over the A A_\infty-operad: e.g. the free resolution in the context of dg-operads of the linear associative operad.

Examples

References

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

For a short and precise introduction see

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

  • B. Keller, A 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 A_\infty-categories is a recent book

  • Yu. Bespalov, V. Lyubashenko, O. Manzyuk, Pretriangulated A 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 A_\infty functors, talk at Categories in geometry and math. physics, Split 2007, slides, pdf, work with Volodymyr Lyubashenko

The relation of A 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 A_\infty-categories, see the overview on p. 3

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

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

For A A_\infty-categories in the wider sense

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

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

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

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

and

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

Revised on April 24, 2013 01:15:07 by Urs Schreiber (131.174.42.75)