The Yoneda isomorphism commutes with homology

Abstract

Abstract. We show that, for a right exact functor from an abelian category to abelian groups, Yoneda's isomorphism commutes with homology and, hence, with functor derivation. Then we extend this result to semiabelian domains. An interpretation in terms of satellites and higher central extensions follows. As an application, we develop semiabelian (higher) torsion theories and the associated theory of (higher) universal (central) extensions.

Higher extensions in exact Mal'tsev categories: distributivity of congruences and the $3^n$-Lemma

“…The aim of this article is to explain how to continue this process: to describe what is a 3 ˆ3 ˆ3-diagram-in some sense, a short exact sequence between 3 ˆ3-diagrams-and so on; see Figure 2 This leads to a 3 n -Lemma for each n ě 2, which extends the 3 ˆ3-Lemma of [2] to higher degrees. We shall see that, in a semi-abelian category [15], the concept of a higher (cubic) extension (in the sense of [10,9,25,24] and the papers referred to there) is equivalent to the notion of a 3 n -diagram introduced here. These may be understood as a non-abelian version of the concept of a Yoneda extension [26], which is useful for instance when studying cohomology of non-abelian algebraic structures [25,24].…”

“…We shall see that, in a semi-abelian category [15], the concept of a higher (cubic) extension (in the sense of [10,9,25,24] and the papers referred to there) is equivalent to the notion of a 3 n -diagram introduced here. These may be understood as a non-abelian version of the concept of a Yoneda extension [26], which is useful for instance when studying cohomology of non-abelian algebraic structures [25,24].…”

“…We recall the inductive definition of a higher cubic extension. (Our terminology here follows [24], where in accordance with [26], "extension" means "short exact sequence" rather than "regular epimorphism"; we here call "cubic extension" what is called "extension" in [10,9] and elsewhere.) Denoting by E the class of cubic extensions in X, a 2-cubic extension (also called a double extension) in X is a commutative square as in Figure 5 where the morphisms a, b, f 1 , f 0 and the universally induced morphism xa, f 1 y : A 1 Ñ A 0 ˆB0 B 1 to the pullback of b and f 0 are in E .…”

“…An 3 n -diagram or n-extension [24,25] under A and over X is an n-sequence E such that each composite below is a short exact sequence:…”

“…In the context of an abelian category, the two are equivalent via (a truncated version of) the Dold-Kan correspondence [7,9]. In a semi-abelian context, 3 n -diagrams occur in the interpretation of the derived functors of Homp´, Aq : X op Ñ Ab for any abelian object A in X; see [25,24]. 5.2.…”