The orbifold/condensation completion procedure of defect topological quantum field theories can be seen as carrying out a lattice or state sum model construction internal to an ambient theory. In this paper, we propose a conceptual algebraic description of orbifolds/condensations for arbitrary tangential structures in terms of higher dagger structures and higher idempotents. In particular, we obtain (oriented) orbifold completion from (framed) condensation completion by using a general strictification procedure for higher dagger structures which we describe explicitly in low dimensions; we also discuss the spin and unoriented case. We provide several examples of higher dagger categories, such as those associated to state sum models, (orbifolds of) Landau--Ginzburg models, and truncated affine Rozansky-Witten models. We also explain how their higher dagger structures are naturally induced from rigid symmetric monoidal structures, recontextualizing and extending results from the literature.
Keywords: Orbifold completion, higher dagger structure, categorified idempotent
2020 MSC: 18N10,18M40,18N25
Theory and Applications of Categories, Vol. 45, 2026, No. 22, pp 863-923.
Published 2026-04-27.
TAC Home