Why the centre is the way it is
Sjoerd Crans – 11 September 1996
1. what is universal property of center of group? Know that dividing by commutator subgroup gives largest quotient that is commutative; does largest commutative subgroup exist? No, because union of commutative groups is not commutative in general. Of course, this is nothing but saying that the inclusion of commutative groups into groups has a left adjoint, but not a right adjoint.
2. observing a group is a category with one object, recognize an element of the center as a natural transformation id -> id. want to use this as basis for general definition of center.
3. short recall of basics of k-monoidal omega-teisi. q-transformations as maps of degree q and as functors C tensor 2_q -> D. Formal reindexing via Omega and Sigma.
4. for k-monoidal omega-tas C, define its k-center by Z_k (C) = Omega^k (omega-Teisi (Sigma^k C, Sigma^k C)). Low dimensional elements are identity transformations, higher dimensional elements are transformations between those.
5. compare this with known definition of center of a monoidal category, where an object of the center is a pair (A, R_A,-): canonical projection Z_k (C) -> C by evaluation at the unique object of Sigma^k C, which explains the A, and q-transformation Sigma^k C tensor 2_q -> Sigma^k C induces, via a quite complicated construction called ``localization of transformations'', a q-transformation C tensor 2_q C, which explains the R_A,-. This gives description of elements of k-center of C as element of C + transformation.
6. This will be sections 2.5.5 and 2.7 of the forthcoming 80+ page paper "Central observations on \omega-teisi".