turkmath.org

Monoids are algebraic devices for describing non-symmetric processes. Actions of monoids receive more attention recently due to applications on various different areas, including automata theory, dynamical systems, quantum computations and data analysis. Unlike group actions, it is particularly difficult to obtain classification results for monoid actions, as most of the standard theorems for groups do not apply to monoids. In this talk, we will discuss how we overcome this issue by using a functor that assigns a collection of symmetries to a non-symmetry. More precisely, to a given space A on which a monoid M acts, we assign a collection spaces on which either M or one of its submonoids acts by homeomorphisms. We use this construction to induce a family of model structures on the category of M-spaces, which allows us to do homotopy theory by using the standard equivariant homotopy theory. We will also discuss some applications of our constructions.