More on internal categories in the operadic context

Steve Lack – 19 November 2003

This talk is based on joint work with Simona Paoli. For a complete and cocomplete symmetric monoidal closed category V, write Op(V) for the category of (symmetric) operads in V. There is a symmetric monoidal closed on the category Cat(V) of internal categories in V, with the closed structure defined in the usual way (with "internal functor categories"). Thus it makes sense to talk about operads in Cat(V). The main result of the talk is the equivalence of categories Cat(Op(V))=Op(Cat(V)). In the special case where V is abelian, there is an equivalence Cat(V)=V2 with the category of morphisms in V, and so Cat(Op(V))=Op(V2). In the further special case where V is the category of graded modules overy a commutative ring, the operads in Op(V2) were studied in a recent preprint of Baues, Minian, and Richter, and called secondary operads. We offer the characterization in terms of internal categories in Op(V) as a conceptually simpler point of view. The second result of the talk concerns the case of an operad T in V, seens as a discrete internal category cat(T) in Op(V); by the first result we may also regard cat(T) as an operad in Cat(V). We show that cat(T)-Alg=Cat(T-Alg), so that internal categories in the category T-Alg of T-algebras can be seen as algebras for the operad cat(T)-Alg. In the context of Baues, Minian, and Richter, there is an operad (0,T) in V2 whose algebras are defined to be crossed modules of T-algebras. This operad corresponds under the equivalence Op(Cat(V))=Op(V2) to cat(T). Thus our result provides further justification for this name by showing that crossed modules are equivalent to internal categories.