A comparison between $SL_n$ spider categories

Abstract We prove a conjecture of Lê and Sikora by providing a comparison between various existing $SL_n$ skein theories. While doing so, we show that the full subcategory of the spider category, $\mathcal {S}p(SL_n)$ , 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 with the CKM and Sikora’s webs are isomorphic.