  A quandle is an algebraic structure which attempts to generalize group
conjugation. These structures have been studied extensively due to their
connections with knot theory, algebraic combinatorics, and other fields. In
this work, we approach the study of quandles from the perspective of the
representation theory of categories. Namely, we look at collections of
conjugacy classes of the symmetric groups and the finite general linear groups,
and prove that they carry the structure of FI-quandles (resp.
VIC$(q)$-quandles). As applications, we prove statements about the homology of
these quandles, and construct FI-module and VIC$(q)$-module invariants of
links.
