Centers and homotopy centers in enriched monoidal categories

We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. The duoidal categories (introduced by Aguiar and Mahajan under the name 2-monoidal categories) are categories with two monoidal structures which are related by some, not necessary invertible, coherence morphisms. Centers of monoids in this sense include many examples which are not ‘classical.’ In particular, the 2-category of categories is an example of a center in our sense. Examples of homotopy center (analogue of the classical Hochschild complex) include the Gray-category Gray of 2-categories, 2-functors and pseudonatural transformations and Tamarkin’s homotopy 2-category of dgdg-categories, dgdg-functors and coherent dgdg-transformations.