The abstract cotangent complex and Quillen cohomology of enriched categories

In his fundamental work, Quillen developed the theory of the cotangent complex as a universal abelian derived invariant, and used it to define and study a canonical form of cohomology, encompassing many known cohomology theories. Additional cohomology theories, such as generalized cohomology of spaces and topological Andr\'e-Quillen cohomology, can be accommodated by considering a spectral version of the cotangent complex. Recent work of Lurie established a comprehensive $\infty$-categorical analogue of the cotangent complex formalism using stabilization of $\infty$-categories. In this paper we study the spectral cotangent complex while working in Quillen's model categorical setting. Our main result gives new and explicit computations of the cotangent complex and Quillen cohomology of enriched categories. For this we make essential use of previous work, which identifies the tangent categories of operadic algebras in unstable model categories. In particular, we present the cotangent complex of an $\infty$-category as a spectrum valued functor on its twisted arrow category, and consider the associated obstruction theory in some examples of interest

On straightening for Segal spaces

The straightening-unstraightening correspondence of Grothendieck--Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$-categories and diagrams of $(\infty, 1)$-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening-unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories

Minimal fibrations of dendroidal sets

We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for ∞–operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. We also explain how our arguments can be used to extend the results of Cisinski (2014) and give the existence of minimal fibrations in model categories of presheaves over generalized Reedy categories of a rather common type. Besides some applications to the theory of algebras over ∞–operads, we also prove a gluing result for parametrized connective spectra (or Γ–spaces)

PD operads and explicit partition lie algebras

Infinitesimal deformations are governed by partition Lie algebras. In characteristic 0, these higher categorical structures are modelled by differential graded Lie algebras, but in characteristic p, they are more subtle. We give explicit models for partition Lie algebras over general coherent rings, both in the setting of spectral and derived algebraic geometry. For the spectral case, we refine operadic Koszul duality to a functor from operads to divided power operads, by taking ‘refined linear duals’ of Σn-representations. The derived case requires a further refinement of Koszul duality to a more genuine setting

Orthofibrations and monoidal adjunctions

We study various types of fibrations over a product of two $\infty$-categories, and show how they can be dualised over one of the two factors via an explicit construction in terms of spans. Among other things, we use this to prove that given an adjunction between monoidal $\infty$-categories, there is an equivalence between lax monoidal structures on the right adjoint and oplax monoidal structures on the left adjoint functor.Comment: 48 page