# Web Categories

## Meeting Details

Abstract: In https://arxiv.org/abs/q-alg/9712003, Kuperberg defines so-called "spiders" for $A_2, B_2$ and $G_2$, generalizing a classical construction of Rumer-Teller-Weyl for $\mathfrak{sl}_2(\mathbb{C})$. These spiders yield a diagrammatic generators and relations description for the monoidal category $\mathbf{Rep}(\mathfrak{g})$ of finite-dimensional complex linear representations of $\mathfrak{g}$. In these diagrammatic categories, hom-spaces consist of braid-like diagrams: so, such categories have come to be called "web" categories. I'll discuss some examples of web categories, mainly focusing on the Temperley-Lieb category $\mathcal{TL}(-2)$ and the partition category. I'll also discuss open problems we are working on such as constructing webs for the finite group $SL_2(\mathbb{F}_q)$ (joint with Daniel Tubbenhauer).