Presenting the Sierpinski gasket in various categories of metric spaces (Q6600687)

This paper provides new presentations of the Sierpinski gasket in terms of universal properties, namely as final coalgebras for functors on three categories of metric spaces with additional designated points, called tripointed metric spaces.\N\NA tripointed metric space \((X,d)\) is a tripointed set (i.e. a tuple \((X, T , L, R)\) where \(X\) is a set along with three distinguished distinct elements \( T , L, R\)) endowed with a 1-bounded metric \(d\) (i.e., \(d(x, y) \le 1\) for every \(x,y\in X\)). The three categories are the following:\N\begin{itemize}\N\item[1.] \(\mathbf{Met}_{\mathbf 3}^{Sh}\): tripointed metric spaces with non-increasing maps which preserve the designated points \(T\), \(L\), and \(R\).\N\item[2.] \(\mathbf{Met}_{\mathbf 3}^{L}\): tripointed metric spaces with Lipschitz maps which preserve the designated points \(T\), \(L\), and \(R\).\N\item[3.] \(\mathbf{Met}_{\mathbf 3}^{C}\): tripointed metric spaces with continuous maps which preserve the designated points \(T\), \(L\), and \(R\).\N\end{itemize}\NThe paper discusses three endofunctors \(F \colon \mathcal C \to \mathcal C\), where \(\mathcal C\) is one of these three categories. Initial \(F\)-algebras and final \(F\)-coalgebras are studied in detail for each \(\mathcal C\).