n-category = (n,n)-category
n-groupoid = (n,0)-category
The “A” is for Associative and the ”” indicates that associativity is relaxed up to higher homotopies without bound on the degree of the homotopies.
If the composition in the linear -category does happen to be strictly associative 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 the Kontsevich–Soibelman reference below.
A category such that
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 .
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
-algebras are the -categories with one object.
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.
For a short and precise introduction see
and for a Fukaya category-oriented introduction see chapter 1 in
A very detailed treatment of -categories is a recent book
Yu. Bespalov, V. 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
Kenji Lefèvre-Hasegawa, Sur les A-infini catégories (arXiv:math/0310337)
If one understands -category as “operadically defined higher category”, then relevant references would include:
See also the references at model structure on algebras over an operad.