Zigzags and free adjunctions
Lorenzo Riva, Martina Rovelli
TL;DR
The paper addresses the problem of freely adjoining right adjoints to morphisms in an $(\infty,1)$-category by introducing a concrete combinatorial model based on zigzags. It constructs the zigzagification functor $\mathcal{Z}_{+}^2$ (and its higher-dimensional generalizations $\mathcal{Z}_{+}^{n+1}$), providing a universal property: any functor from a Segal space $X$ to a sinister $(\infty,2)$-category $\mathscr{D}$ factors uniquely through $\mathcal{Z}_{+}^2(X)$, thereby ensuring every $1$-morphism gains a right adjoint and the snake equations hold via explicit unit/counit data. The work develops a robust framework using higher Segal spaces, globularity, and a higher square functor $\mathrm{Sq}^n$, culminating in a detailed 2-dimensional universal property and a clear strategy to extend to higher dimensions, with speculative connections to the cobordism hypothesis and oriented tangles. These constructions provide a principled, combinatorial path toward understanding adjunctions in higher categories and potentially linking them to cobordism-type invariants. The results offer explicit generators and a universal mapping property crucial for embedding adjunction data into higher categorical contexts.
Abstract
We construct an explicit combinatorial model of the functor which adds right adjoints to the morphisms of an $\infty$-category, and we speculate on possible extensions to higher dimensions.