We take a novel lattice-theoretic approach to the $\tau$-cluster morphism category $\mathfrak{ T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{ tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: \mathfrak{ T}(A) \to \mathfrak{ T}(A/I)$. If $\mathrm{ tors } A$ is finite, $F_I$ is a regular epimorphism in the category of small categories and we characterise when $F_I$ is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the $\tau$-cluster morphism category up to equivalence.
Categories: math.RT