Deformation Theory for $(\infty,n)$-categories
Roman Kositsyn
TL;DR
This work develops a deformation-theoretic framework for $(\infty,n)$-categories by constructing a twisted arrow category $\mathrm{TwAr}(\mathscr E)$ and proving the central stabilization theorem $\mathrm{Stab}(\mathrm{Cat}_{n,/\mathscr E}) \cong \mathrm{Hom}_{\mathrm{Cat}}(\mathrm{TwAr}(\mathscr E), \mathrm{Sp})$. It introduces Steiner complexes to handle complex pasting diagrams, and uses $\Theta_n$-trees to manage combinatorics of pasting in higher dimensions. An alternative, computable model for stabilization is provided via $\mathrm{TwAr}_\theta(\mathscr E)$, with explicit computations for Stein objects and a DK-type equivalence guiding the DK-triple perspective. The paper then ties these constructions to a structural description of lax-idempotent monads in the $\infty$-categorical setting, showing deformation-theoretic criteria suffice to characterize such monads and connecting to walking comonads. Overall, it offers a coherent deformation-theory toolkit for $(\infty,n)$-categories and their monadic structures, with concrete models and comparisons across multiple higher-categorical frameworks.
Abstract
For an $(\infty,n)$-category $\mathscr E$ we define an $(\infty,1)$ category $\mathrm{TwAr}(\mathscr E)$ and provide an isomorphism between the stabilization of the overcategory of $\mathscr E$ in $\mathrm{Cat}_{(\infty,n)}$ and the $\infty$-category of spectrum-valued functors on $\mathrm{TwAr}(\mathscr E)$. We use this to develop the deformation theory of $(\infty,n)$-categories and apply it to given an $\infty$-categorical characterization of lax-idempotent monads.