A Crystal Temperley—Lieb Category

By considering a suitable renormalization of the Temperley--Lieb category, we study its specialization to the case $q=0$. Unlike the $q\ne 0$ case, the obtained monoidal category, $TL_0(\Bbbk)$, is not rigid or braided. We provide a closed formula for the Jones--Wenzl projectors in $TL_0(\Bbbk)$ and give semisimple bases for its endomorphism algebras. We then describe a coboundary structure on $TL_0(\Bbbk)$ and show that its idempotent completion is coboundary monoidally equivalent to the category of $\mathfrak{sl}_2$-crystals. This gives a diagrammatic description of the commutor for $\mathfrak{sl}_2$-crystals defined by Henriques and Kamnitzer and of the resulting action of the cactus group. Joint work with Mateusz Stroiński.