Factorizations of surjective maps of connected quandles

We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. (2008) and the factorization system for surjective quandle homomorphsims of Bunch et al. (2010) as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was supported by National Science Foundation, grant DMS-1659123.