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We remark on the claim that the string-net model of Levin and Wen is amicroscopic Hamiltonian formulation of the Turaev-Viro topological quantumfield theory. Using simple counterexamples we indicate where interesting extrastructure may be needed in the Levin-Wen model for this to hold (however webelieve that some form of the correspondence is true). In order to beaccessible to the condensed matter community we provide a brief and gentleintroduction to the relevant concepts in category theory (relying heavily onanalogy with ordinary group representation theory). Likewise, some physicalideas are briefly surveyed for the benefit of the more mathematical reader. Themain feature of group categories under consideration is Turaev's unimodality.We pinpoint where unimodality should fit into the Levin-Wen construction, andshow that the simplest example fails to be unimodal. Unimodality isstraightforward to compute for group categories, and we provide a completeclassification at the end of the paper.

Counterexamples in Levin-Wen string-net models, group categories, and Turaev unimodality