turkmath.org

In this talk we will discuss the definitions of (co)homology and (co)homotopy theories graded over a monoidal category by using bi-actions of monoidal categories on categories. Later, we will show that these defintions agree with the existing definitions of (co)homology and (co)homotopy theories for particular choices of monoidal categories and their actions. Moreover we will define $\Omega$-spectra and suspension spectra as fixed points of certain monoidal actions and show how these fixed points can represent (co)homology and (co)homotopy theories. This talk is based on a joint work with Özgün Ünlü.