A higher categorical approach to the André-Quillen cohomology of an (∞, 1)-Category

Abstract: Simplicial categories, that is categories enriched in simplicial sets, are a model of (∞, 1)-categories. Their André-Quillen cohomology, originally introduced by Dwyer, Kan and Smith [DKS], was later re-interpreted and extended by Harpaz, Nuiten and Prasma [HNP1]. The André-Quillen cohomology of a simplicial category can be used to describe its k-invariants which in turn contain various higher homotopy information and in particular yield an obstruction theory for realizing homotopy-commutative diagrams [DKS]. Our aim is to give an algebraic, elementary and explicit approach to the André-Quillen cohomology of simplicial categories using the tools of higher category theory.

For this purpose, we first observe that in order to study the nth André-Quillen cohomology group of a simplicial category, it suffices to look at simplicial categories that are n-truncated, that is they are enriched in n-types. This has the advantage that we can use one of the algebraic models of n-types from higher category theory to produce an algebraic replacement for the nth Postnikov truncation of a simplicial category. We choose to use the category of groupoidal weakly globular n-fold categories arising within Paoli's model of weak n-categories [Pa3]. This category is a model of n-types with a cartesian monoidal structure. Further, every n-type can be modelled by a weakly globular n-fold groupoid, that is an object of the full subcategory of weakly globular n-fold groupoids [BP2], which is more convenient algebraically. Our model for the nth Postnikov truncation of a simplicial category is a category enriched in weakly globular n-fold groupoids with respect to the cartesian monoidal structure. We call the latter an n-track category. Using the n-fold nature of our model, we iteratively build a comonad on n-track categories. Using this comonad we then obtain an explicit cosimplicial abelian group model for the André-Quillen cohomology of an (∞, 1)-category. This is joint work with David Blanc [BP4].

References:

[BP2] D. Blanc & S. Paoli, Segal-type algebraic models of n-types, Algebraic & Geometric Topology 14 (2014), pp. 3419-3491.

[BP4] D. Blanc & S. Paoli, A Model for the André-Quillen Cohomology of an (∞, 1)-Category, preprint arXiv:2405.12674v2, 2024.

[DKS] W.G. Dwyer, D.M. Kan, J. H. Smith An obstruction theory for diagrams of simplicial categories, Proc.Kon. Ned. Akad. Wet. - Ind. Math. 48 (1986), pp. 153-161.

[HNP1] Y. Harpaz, J. Nuiten, & M. Prasma, The abstract cotangent complex and Quillen cohomology of enriched categories, J. Topology 11 (2018), 752-798.

[Pa3] S. Paoli, Simplicial Methods for Higher Categories: Segal-type models of weak n-categories, 'Algebra and Applications', Springer, Berlin-New York, 2019.

commutative algebraalgebraic geometryalgebraic topologycategory theorygeometric topologyK-theory and homology

Audience: researchers in the topic

Comments: Join Zoom Meeting unipv-it.zoom.us/j/94344875868 Meeting ID: 943 4487 5868

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Transalpine Topology Tetrahedron (TTT) - Pavia Vertex

Series comments: The Transalpine Topology Tetrahedron (TTT) is a topology seminar partially supported by the London Mathematical Society (LMS) and INdAM. It has UK nodes at Liverpool, Sheffield and Warwick and an international node at Pavia. For many years TTT stood for Transpennine Topology Triangle.

Website: sarah-whitehouse.sites.sheffield.ac.uk/transalpine-topology-tetrahedron

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