Representation of n-abelian categories in abelian categories

Abstract

We prove a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categories. More precisely, for a positive integer n and a small n-abelian category M, we show that M is equivalent to a full subcategory of an abelian category L2(M,G), where L2(M,G) is the category of absolutely pure group valued functors over M. We also show that n-kernels and n-cokernels in M are precisely exact sequences of L2(M,G) with terms in M.