{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T18:20:10Z","timestamp":1758478810226,"version":"3.40.5"},"reference-count":52,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2018,12,4]],"date-time":"2018-12-04T00:00:00Z","timestamp":1543881600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Inst. Math. Jussieu"],"published-print":{"date-parts":[[2020,11]]},"abstract":"<jats:p>The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S1474748018000555_inline1.png\"\/><jats:tex-math>${\\mathcal{J}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>of a fiat 2-category<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S1474748018000555_inline2.png\"\/><jats:tex-math>$\\mathscr{C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S1474748018000555_inline3.png\"\/><jats:tex-math>$\\mathscr{C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>associated with a diagonal<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S1474748018000555_inline4.png\"\/><jats:tex-math>${\\mathcal{H}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-cell in<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S1474748018000555_inline5.png\"\/><jats:tex-math>${\\mathcal{J}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. 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