Homology, Homotopy and Applications

Crossed modules of racks

Alissa S. Crans and Friedrich Wagemann

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We generalize the notion of a crossed module of groups to that of a crossed module of racks. We investigate the relation to categorified racks, namely strict 2-racks, and trunk-like objects in the category of racks, generalizing the relation between crossed modules of groups and strict 2-groups. Then we explore topological applications. We show that by applying the rackspace functor, a crossed module of racks gives rise to a covering. Our main result shows how the fundamental racks associated to links upstairs and downstairs in a covering fit together to form a crossed module of racks.

Article information

Homology Homotopy Appl., Volume 16, Number 2 (2014), 85-106.

First available in Project Euclid: 22 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) [See also 20Axx, 20L05, 20Mxx] 18D05: Double categories, 2-categories, bicategories and generalizations 18D35: Structured objects in a category (group objects, etc.) 20L05: Groupoids (i.e. small categories in which all morphisms are isomorphisms) {For sets with a single binary operation, see 20N02; for topological groupoids, see 22A22, 58H05} 57M10: Covering spaces 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}

Crossed module of groups rack augmented rack crossed module of racks rack space covering of rack space link invariant fundamental rack of a link 2-group 2-rack


Crans, Alissa S.; Wagemann, Friedrich. Crossed modules of racks. Homology Homotopy Appl. 16 (2014), no. 2, 85--106. https://projecteuclid.org/euclid.hha/1408712336

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