(also nonabelian homological algebra)
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An -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 “” 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 -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 -category happen to be equal to identity morphisms (which is encoded by the vanishing of the maps for , defined below), it becomes the same as a dg-category. In fact, every linear -category is -equivalent to a dg-category. In this way, we have that -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.
In what is strictly speaking a restrictive sense – which is however widely and conventionally understood in homological algebra as the standard notion of -category (see references below) – the hom-spaces of an -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 -category in this standard sense of homological algebra is a category which is in some way homotopically enriched over a category of chain complexes . Since a category which is enriched in the ordinary sense of enriched category theory is a dg-category, there is a close relation between -categories and dg-categories.
-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.
A category such that
for all in the Hom-sets are finite dimensional chain complexes of -graded modules
for all objects in there is a family of linear composition maps (the higher compositions) of degree (homological grading convention is used) for
is the differential on the chain complex
satisfy the quadratic -associativity equation for all .
and will be chain maps but the compositions 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 -categories and -functors. Many features of -categories and -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 -categories are weak -categories with all morphisms invertible. -categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.
Every dg-category may be regarded as a special case when there are no higher maps (trivial homotopies) of an -category.
Every -category is -equivalent to a dg-category.
This is a corollary of the -categorical Yoneda lemma.
beware that this statement does not imply that the notion of -categories is obsolete (see section 1.8 in Bespalov et al.): in practice it is often easier to work with a given naturally arising -category than constructing its equivalent dg-category
for instance when dealing with a Fukaya? -category;
or when dealing with various constructions on dg-categories, for instance certain quotients,that naturally yield directly -categories instead of dg-categories.
The path space of a topological space
The Fukaya category of a topological space – a Calabi-Yau A-∞ category
-algebras are the -categories with one object.
In the widest sense, -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 .
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.
An -category is a category over the -operad: e.g. the free resolution in the context of dg-operads of the linear associative operad.
also the classical notion of bicategory can be interpreted as an -category in Cat for a suitable Cat-operad.
For a short and precise introduction see
B. Keller, Introduction to -algebras and modules (dvi, ps) and Addendum (ps), Homology, Homotopy and Applications 3 (2001), 1-35;
B. Keller, algebras, modules and functor categories, (pdf, ps).
and for a Fukaya category-oriented introduction see chapter 1 in
A very detailed treatment of -categories is a recent book
Yu. Bespalov, Volodymyr Lyubashenko, O. Manzyuk, Pretriangulated -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)
Oleksandr Manzyk, A-infinity-bimodules and Serre A-infinity-functors, dissertation pdf, djvu; Serre functors, talk at Categories in geometry and math. physics, Split 2007, slides, pdf, work with Volodymyr Lyubashenko
The relation of -categories to differential graded algebras is emphasized in the introduction of
which mostly discusses just A-infinity-algebras, but points out a generalizations to -categories, see the overview on p. 3
Essentially the authors say that an -category should be a non(-graded-)commutative dg-manifold/L-infinity-algebroid.
More category theory and homotopy theory of -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
If one understands -category as “operadically defined higher category”, then relevant references would include:
With operads modeled by dendroidal sets, n-categories for low viewed as objects with an -composition operation are discussed in section 5 of
and
See also the references at model structure on algebras over an operad.
Last revised on October 18, 2022 at 11:15:36. See the history of this page for a list of all contributions to it.