Bisets as categories, and tensor product of induced bimodules

Bisets can be considered as categories. This note uses this point of view to give a simple proof of a Mackey-like formula expressing the tensor product of two induced bimodules.

💡 Research Summary

The paper “Bisets as categories, and tensor product of induced bimodules” presents a concise categorical framework for bisets and uses it to derive a Mackey‑type decomposition formula for the tensor product of two induced bimodules. The authors begin by observing that a (G, H)‑biset X can be regarded as a small category C_X: the objects are the elements of X, and a morphism from x to y is a pair (g, h)∈G×H such that g·x·h⁻¹ = y. In this category the automorphism group of an object x is precisely the stabilizer Stab_{G×H}(x). This viewpoint allows one to treat any k‑linear (G, H)‑bimodule M as a functor F_M:C_X→Mod_k by assigning to each object x the submodule of M fixed by Stab_{G×H}(x) and defining the action of a morphism (g, h) via the natural G‑ and H‑actions.

With this functorial description the usual induced bimodule Ind_X^G M is identified with a coend (or colimit) over C_X: \