A comparison between $S{L}_n$ spider categories

We prove a conjecture of Lê and Sikora by providing a comparison between various existing $S{L}_n$ skein theories. While doing so, we show that the full subcategory of the spider category, $\mathcal{S}p(S{L}_n)$, defined by Cautis-Kamnitzer-Morrison, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora’s quotient category. This also answers a question from Morrison’s Ph.D. thesis. Finally, we show that the skein modules associated to the CKM and Sikora’s webs are isomorphic.