Український математичний конгрес - 2009
Володимир Любашенко (Інститут математики НАН України, Київ, Україна)
A∞-categories via operads and multicategories
A∞-algebras and (unital) A∞-categories are generalizations of dg-algebras and dg-categories. They are related to operads and multicategories in two ways. First of all, operations in (unital) A∞-algebras and A∞-categories form a (non-symmetric) dg-operad, which is an instance of enriched multicategory with one object. This dg-operad is a resolution of the corresponding notion for dg-algebras. We discuss various notions of unitality for A∞-categories and their equivalence in non-filtered case. Secondly, A∞-categories form a closed category and, moreover, they are objects of a symmetric closed multicategory. The latter property holds also for unital A∞-categories. This point of view leads to the notion of a pretriangulated A∞ -category, which is a generalization of a pretriangulated dg-category.