69 research outputs found
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 -categorical analogue of the cotangent
complex formalism using stabilization of -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 -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 -categories and diagrams of -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
-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
-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
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