Tag - Model categories

Carles Casacuberta: Homotopy reflectivity is equivalent to the weak Vopenka principle

21st March, 2024

We discuss reflectivity of colocalizing subcategories of triangulated categories under suitable set-theoretical assumptions. In earlier joint work with Gutierrez and Rosicky, we proved that if K is any locally presentable category with a stable model category structure, then Vopenka's principle implies that every full subcategory L of the homotopy category of K closed under products and fibres is reflective. Moreover, if L is colocalizing, then the reflection is exact. Using recent progress in large-cardinal theory, we show that the statement that every full subcategory closed under products and fibres is reflective is, in fact, equivalent to the so-called weak Vopenka principle. Hence this statement cannot be proved using only the ZFC axioms.

Matt Booth: Global Koszul duality

7th March, 2024

Conilpotent Koszul duality, as formulated by Positselski and Lefevre-Hasegawa, gives an equivalence (of model categories, or of ∞-categories) between augmented dg algebras and conilpotent dg-coalgebras. One should think of this as a non-commutative version of the Lurie-Pridham correspondence: indeed in characteristic zero, cocommutative conilpotent dg coalgebras are Koszul dual to dg Lie algebras, and this is precisely the correspondence between formal moduli problems and their tangent complexes. I'll talk about a global analogue where the conilpotency assumption is removed; geometrically this corresponds to non-commutative formal moduli problems modelled on profinite completions, rather than pro-Artinian completions. Global Koszul duality is best expressed as a Quillen equivalence between curved dg algebras and curved dg coalgebras, and in both categories the weak equivalences are defined using an auxiliary object, the Maurer-Cartan dg category of a curved dg algebra.

Leovigildo Alonso Tarrio: Derivators in additive context

20th October, 2023

By a theorem of Cisinksi, every combinatorial model category defines a strong derivator. For a Grothendieck category A, there are several combinatorial model structures defined on A, thus its derived category is the base of a strong derivator. In this talk, we present an alternative path to this result assuming further that A has enough projective objects. This approach has the benefit of simplicity (and less prerequisites) and gives a very explicit description of homotopy Kan extensions, in particular homotopy limits and colimits. We will present these results. Further, as an application, we will show how to extend the description of local cohomology via Koszul complexes from closed subsets to arbitrary systems of supports, i.e. stable for specialization subsets. Time permitting, we will discuss how this point of view applies to the co/homology of groups.