Mackey 2-Functors and Mackey 2-Motives

Abstract

We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [BD20], obtained by modding out the so-called cohomological relations. This categorifies Yoshida's Theorem for ordinary cohomological Mackey functors, and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

On Krull-Schmidt bicategories

“…Examples of Krull-Schmidt bicategories are discussed in Section 7. They include various bimodule bicategories of rings (such as rings with noetherian center), bicategories of Mackey 2-motives [BD20] (our original motivation for this note), semisimple 2-categories such as that of finite dimensional 2-vector spaces [KV94] or 2-Hilbert spaces [Bae97], as well as the 2-category of finite dimensional 2-representations of an arbitrary 2-group over an arbitrary field (Corollary 7.15). Generalizing the last example, we also categorify the result that finite dimensional representations of any algebra form a Krull-Schmidt category (Theorem 7.12).…”

“…we do not feel any pressing need to discuss higher versions of bi-chain conditions, finite length objects, etc. In order to recognize a Krull-Schmidt bicategory using our characterization, it suffices to ensure weak block completion (which can always be implemented by [BD20,Thm. A.7.23]) and to have a reasonable grasp of the 2-cell Homs (e.g.…”

“…An additive bicategory is weakly block complete if for every object X and every idempotent element e = e 2 of the ring End B (Id X ) there exists a direct sum decomposition X ≃ X 1 ⊕ X 2 under which the 2-cells e and 1 X − e correspond to [ 1 0 0 0 ] and [ 0 0 0 1 ], respectively, as in Remark 2.8. As in [BD20], we say that B is block complete if moreover every Hom category of B is idempotent complete. 2.11.…”

“…2.11. Theorem (See [BD20,). For every additive bicategory B, there exists a block complete bicategory B ♭ and a 2-fully faithful pseudofunctor ι : B ֒→ B ♭ which is 2-universal among additive pseudofunctors to block complete bicategories.…”

“…Let k be any noetherian commutative ring. Every bicategory of klinear Mackey 2-motives in the sense of [BD20] is Krull-Schmidt. Similarly, every bicategory of cohomological Mackey 2-motives as in [BD21] is Krull-Schmidt.…”