Orbifold Completion of 3-Categories

Abstract We develop a general theory of 3-dimensional “orbifold completion”, to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. Given a semistrict 3-category $$\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> with adjoints for all 1- and 2-morphisms (more precisely, a Gray category with duals), we construct the 3-category $${\mathcal {T}}_{\text {orb}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> </mml:math> as a Morita category of certain $$E_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> -algebras in $$\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> which encode triangulation invariance. We prove that in $${\mathcal {T}}_{\text {orb}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> </mml:math> again all 1- and 2-morphisms have adjoints, that it contains $$\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> as a full subcategory, and we argue, but do not prove, that it satisfies a universal property which implies $${({\mathcal {T}}_{\text {orb}})}_{\text {orb}} \cong {\mathcal {T}}_{\text {orb}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>orb</mml:mtext> </mml:msub> <mml:mo>≅</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> </mml:mrow> </mml:math> . This is a categorification of the work in Carquevill and Runkel (Quantum Topol 7(2):203–279, 2016). Orbifold completion by design allows us to lift the orbifold construction from closed TQFT to the much richer world of defect TQFTs. We illustrate this by constructing a universal 3-dimensional state sum model with all defects from first principles, and we explain how recent work on defects between Witt equivalent Reshetikhin–Turaev theories naturally appears as a special case of orbifold completion.