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We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of$\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projectivezigzag modules (which categorifies the action of the Weyl group $W$ on the rootlattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ inducedby the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on$\mathbb{B}_W$ agrees with the word-length metric in the canonical generatorsof the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We alsodefine, for each choice of a Coxeter element $c$ in $W$, a baric structure on$\mathcal{K}$. We use these baric structures to define metrics on the braidgroup, and we identify these metrics with the word-length metrics in theBirman-Ko-Lee/Bessis dual generators of the associated dual positive monoid$\mathbb{B}_{W.c}^\vee$. As consequences, we give new proofs that the standardand dual positive monoids inject into the group, give linear-algebraicsolutions to the membership problem in the standard and dual positive monoids,and provide new proofs of the faithfulness of the action of $\mathbb{B}_W$ on$\mathcal{K}$. Finally, we use the compatibility of the baric and$t$-structures on $\mathcal{K}$ to prove a conjecture of Digne and Gobetregarding the canonical word-length of the dual simple generators of ADE braidgroups.

Braid groups of type ADE, Garside monoids, and the categorified root lattice