A construction of semisimple tensor categories
Let \(K\) be a field of characteristic zero. The author extends \textit{P. Deligne}'s recent construction of a tensor category \({\mathcal R}ep(S_t,K)\) of representations of the ``symmetric group'' \(S_t\) with \(t\in K\) not necessarily a positive integer. Note that in case \(K\) is algebraically closed and \(t\in\mathbb N\), the tensor category \(\mathcal C_t\) of ordinary representations of \(S_t\) determines \(S_t\) up to isomorphism. In this case, \(\mathcal C_t\) is a specialization of \({\mathcal R}ep(S_t,K)\). For an abelian category \(\mathcal A\), the author constructs a semisimple tensor category \(\mathcal T(\mathcal A,K)\) in several steps. First, he extends the morphisms \(x\rightarrow y\) of \(\mathcal A\) to correspondences, i.e., equicalence classes of maps \(F\colon c\rightarrow x\oplus y\), where \(F\colon c\rightarrow x\oplus y\) and \(G\colon d\rightarrow x\oplus y\) are equivalent if and only if \(\text{Im}\, F=\text{Im}\, G\), and the kernels of \(F\) and \(G\) coincide in the Grothendieck monoid of \(\mathcal A\). The composition of correspondences is defined in the obvious way, like in Deligne's paper. Then the morphisms are extended to \(\mathbb Z\)-linear combinations of correspondences. The biproduct in the category \(\mathcal T_1(\mathcal A)\) thus obtained gives rise to a symmetric monoidal structure with unital object 0. Moreover, \(\mathcal T_1(\mathcal A)\) is \(\mathbb Z[\text{simp}(\mathcal A)]\)-linear, where \(\text{simp}(\mathcal A)\) is a representative system of the isomorphism classes of simple objects in \(\mathcal A\). To make the category \(K\)-linear, a homomorphism \(t\colon\mathbb Z[\text{simp}(\mathcal A)] \rightarrow K\) has to be fixed, such that \(K\) becomes a \(\mathbb Z[\text{simp}(\mathcal A)]\)-module. Finally, the category is made pseudo-abelian (SGA~4), i.e., additive with splitting idempotents, in a canonical way, which gives a category \(\mathcal T(\mathcal A,K)\) with a symmetric \(K\)-bilinear rigid tensorproduct. If every object has only finitely many subobjects, and \(t\) is chosen properly, the author proves that \(\mathcal T(\mathcal A,K)\) is a semisimple tensor category such that the simple objects are given by pairs \((x,\pi)\) with \(x\in\text{Ob}\,\mathcal A\), where \(\pi\) is a \(K\)-linear representation of \(\text{Aut}_\mathcal A(x)\). Apart from the standard example \(\mathcal A=\mathbb F_q{\mathcal M}od\), the author provides other interesting examples, e.~g., if \(\mathcal A\) is the category of finite abelian \(p\)-groups, then \(\mathcal T(\mathcal A,K)\) specializes to the category of \(K\)-linear representations of \(\text{GL}_n(\widehat{\mathbb Z}_p)\) if \(t\) does not take the values \(0,1,p,p^2,\ldots\)\ . The author remarks that Deligne's category \({\mathcal R}ep(S_t,K)\) is obtained by taking for \(\mathcal A\) the opposite of the category of finite sets. This shows that the construction works for non-abelian, even non-additive categories \(\mathcal A\). It seems that some further investigation is necessary to get a generalization which covers this and other non-additive cases mentioned in the paper.
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