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<p>We introduce group-theoretical fusion 2-categories, a categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara–Yamagami defect, that is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5249_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Z}/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>-graded fusion 2-categories whose non-trivially graded factor is <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5249_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{2Vect}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">2</mn> <mi mathvariant="bold">Vect</mi> </mrow> </math></EquationSource> </InlineEquation>. We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara–Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results.</p>

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Fiber 2-Functors and Tambara–Yamagami Fusion 2-Categories

  • 期刊论文

  • 发表日期：

    2025年2月20日

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  • Thibault D. Décoppet,
  • Matthew Yu

摘要

We introduce group-theoretical fusion 2-categories, a categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara–Yamagami defect, that is \(\mathbb {Z}/2\) Z / 2 -graded fusion 2-categories whose non-trivially graded factor is \(\textbf{2Vect}\) 2 Vect . We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara–Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results.