This Transalpine Topology Tetrahedron (TTT) -International Meeting at the Pavia Vertex is partially funded by the INdAM- GNSAGA grant CUP E53C24001950001, the Dipartimento di Matematica ‘Felice Casorati’ of the University of Pavia and the London Mathematical Society (LMS).

Date Thursday 3rd July 2025

Speakers 

Luigi Caputi (University of Bologna)

Nicholas Meadows (University of Bologna)

Paul Arne Østvær (University of Milan)

Simona Paoli (University of Aberdeen)

Schedule

All talks will take place in Aula Beltrami of the Dipartimento di Matematica ‘Felice Casorati’, Università di Pavia and online streaming via ZOOM.  All times are in CEST.

10:15 - 11:15   Simona Paoli (University of Aberdeen)

                               A higher categorical approach to the André-Quillen cohomology of an 

                               (∞, 1)-Category

11:30 - 12:30   Nicholas Meadows (University of Bologna)

                               Andrè-Quillen Cohomology and the k-Invariants of Simplicial Categories

12:30 Lunch Break

15:00 – 16:00   Luigi Caputi (University of Bologna)

                               Gerstenhaber-Schack Hochschild cohomology via cohomology of  

                               categories

16:00 Coffee Break

16:30 – 17:30   Paul Arne Østvær (University of Milano and University of Oslo)

                               Motivic K-theory

19:00 Dinner at local restaurant

Titles and abstracts

Speaker: Luigi Caputi (University of Bologna)

Title: Gerstenhaber-Schack Hochschild cohomology via cohomology of categories

Abstract: In this talk, we revisit and explore the connection between the Gerstenhaber-Schack Hochschild cohomology and the Baues-Wirsching cohomology of categories. In particular, we analyse the behaviour of Hochschild cohomology of diagrams of homological epimorphisms of algebras. As a main application, we compute the Hochschild cohomology of diagrams of incidence algebras associated to finite filtrations of simplicial complexes. This is joint work with F. Vaccarino. 

Speaker: Nicholas Meadows (University of Bologna)

Title: Andrè-Quillen Cohomology and the k-Invariants of Simplicial Categories

Abstract: Spaces, and more generally infinity-categories, have a canonical decomposition into simpler pieces known as Postnikov sections, which are glued together by their k-invariants.  For an $\infty$-category X, these take value in the spectral Andre-Quillen cohomology of Harpaz, Nuiten, and Prasma. By pulling these k-invariants back to diagrams inside X, one obtains a series of obstructions to lifting the diagram to successive stages in the Postnikov tower. 

In this talk, we will show how various constructions in algebraic topology, such as differentials in spectral sequences and cohomology operations, can be viewed as obstructions to extending cubical diagrams in the infinity category of spaces. Motivated by this, we will also show that there exists a canonical cubical decomposition of the spectral Andre-Quillen cohomology. Joint work with David Blanc. 

Speaker: Paul Arne Østvær (University of Milano and University of Oslo)

Title: Motivic K-theory

Abstract: We will discuss definitions and properties of motivic K-theory of motivic ring spectra. Joint work with Hadrian Heine.

Speaker: Simona Paoli (University of Aberdeen)

Title: 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 and Kan [DK2], 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 [DK2]. 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.

[DK2] W.G. Dwyer & D.M. Kan, An obstruction theory for diagrams of simplicial sets, Proc.

Kon. Ned. Akad. Wet. - Ind. Math. 46 (1984), pp. 139-146.

[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.

Many thanks to Frank Neumann for organising TTT122.