Geodesic
Partially multiplicative quandles and simplicial Hurwitz spaces
Documenta mathematica, Tome 30 (2025) no. 3, pp. 611-672

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We introduce partially multiplicative quandles (PMQ), a generalisation of both partial monoids and quandles. We set up the basic theory of PMQs, focusing on the properties of free PMQs and complete PMQs. For a PMQ Q with completion Q​, we introduce the category of Q​-crossed topological spaces, and define the Hurwitz space HurΔ(Q): it is a Q​-crossed space, and it parametrises Q-branched coverings of the plane. The definition recovers classical Hurwitz spaces when Q is a discrete group G. Finally, we analyse the class of PMQs Sdgeo​ arising from the symmetric groups Sd​, and we compute their enveloping groups and their PMQ completions.
DOI : 10.4171/dm/996
Classification : 55R80, 08A05, 08A35, 18M15, 20B05, 20M05
Mots-clés : quandle, partial monoid, Hurwitz space, bar construction, free group, symmetric group 
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     author = {Andrea Bianchi},
     title = {Partially multiplicative quandles and simplicial {Hurwitz} spaces},
     journal = {Documenta mathematica},
     pages = {611--672},
     publisher = {mathdoc},
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     number = {3},
     year = {2025},
     doi = {10.4171/dm/996},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/996/}
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Andrea Bianchi. Partially multiplicative quandles and simplicial Hurwitz spaces. Documenta mathematica, Tome 30 (2025) no. 3, pp. 611-672. doi: 10.4171/dm/996

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