Preprojective categories of type A

We introduce a continuous version of preprojective algebras of type $A$. In particular, we are interested in the preprojective category over an open, bounded subinterval $\mathbb{I}$ of $\mathbb{R}$, denoted $\Lambda_{\mathbb{I}}$. We study the representable projective modules and define a useful type of sub- and quotient module called decorous modules. These are completely described by a function from the closure $\overline{\mathbb{I}}$ of $\mathbb{I}$ to $\mathbb{R}$ whose 'slopes' are not too steep anywhere. We later use these to describe permuton ideals, a generalization of the support $\tau$-tilting ideals of preprojective algebras of type $A_n$, which we call permutation ideals. Once we have our generalization, we show that permutation ideals can be recovered from permuton ideals. Moreover, permutation ideals are $\tau$-rigid and we show an analogous property for our permuton ideals. Along the way, we classify all the brick $\Lambda_{\mathbb{I}}$-modules.