turkmath.org

In this talk I will describe a homotopy theory for actions of monoids that is built by analyzing their ``reversible parts". Let $M$ be a monoid. For each submonoid $N\leq M$ let $G(N)$ be the group completion of $N$. Given an $M$-space $X$ and a submonoid $N\leq M$, we associate a $G(N)$-space $q_*^N(X)$ which sorts out “symmetries” of the $N$-space $X$ with the restricted $N$-action. By using these $q_*^N$'s we induce a model structure on the category of $M$-spaces and $M$-equivariant and show that this model structure is Quillen equivalent to the projective model structure on the category of contravariant $\mathbf{O}(M)$-diagrams of spaces, where $\mathbf{O}(M)$ is the category whose objects are induced orbits $M\times_N G(N)/H$ for each $N\leq M$ and $H\leq G(N)$ and morphisms are $M$-equivariant maps. Finally, if time permits, I will state some applications.