Australian Category Seminar

Let M be the endomorphism monoid of 2 in Cat. Then the category of right M-sets is the category of graphs (directed, with chosen endo-edges at each vertex). Let $\omega = \{0,1,2,\dots\}$ as an ordered set. Call $\alpha \colon \omega \to \omega$ in Cat eventually consecutive when there exists $k$ such that $\alpha(i + 1) = \alpha(i) + 1$ for all $i>k$. Let $\Delta$ denote the monoid of eventually constant $\alpha \colon \omega \to \omega$ under composition. The category of right $\Delta$-sets is essentially the category of simplicial sets: there can be some infinite dimension cells. (This was not the main point of the lecture.) The remainder of the lecture described $O_n$, giving some technicalities as to why no loops occur.