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A quandle is an algebraic structure which attempts to generalize groupconjugation. These structures have been studied extensively due to theirconnections with knot theory, algebraic combinatorics, and other fields. Inthis work, we approach the study of quandles from the perspective of therepresentation theory of categories. Namely, we look at collections ofconjugacy 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 ofthese quandles, and construct FI-module and VIC$(q)$-module invariants oflinks.

Asymptotic behaviors in the homology of symmetric group and finite general linear group quandles