The Cuboid Lemma and Mal’tsev Categories

2-Categories and Yoneda lemma

A∞-categories and the Yoneda lemma

Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A ∞ -functor Y : A → A ∞ (A,C) is a full embedding for an arbitrary unital A ∞ -category A. For a differential graded k-quiver B we define the free A ∞ -category FB generated by B. The main result is that the restriction A ∞ -functor A ∞ (FB,A) → A1(B,A) is an equivalence, where objects of the l...

The Yoneda Lemma for unital A ∞ - categories

Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A∞-functor Y : A op → A∞(A,C) is a full embedding for an arbitrary unital A∞-category A. Since A∞-algebras were introduced by Stasheff [Sta63, II] there existed a possibility to consider A∞-generalizations of categories. It did not happen until A∞-categories were encountered in studies of mirror...

Semidirect Products and Split Short Five Lemma in Normal Categories

The categorical definition of semidirect products was introduced by D. Bourn and G. Janelidze in [2], where they proved that, in the category of groups, this notion coincides with the classical one. A characterization of pointed categories with categorical semidirect products was given in [3]. The existence of such products imply, in particular, that the category is protomodular, i.e. the Split...

Maltsev Families of Varieties Closed under Join or Maltsev Product

Maltsev families of varieties which are closed under join or Maltsev product are investigated. New Maltsev conditions for congruence semi-distributivity are given.