We introduce the notion of M-locally generated category for a factorization system (E,M) and study its properties. We offer a Gabriel-Ulmer duality for these categories, introducing the notion of nest. We develop this theory also from an enriched point of view. We apply this technology to Banach spaces showing that it is equivalent to the category of models of the nest of finite-dimensional Banach spaces.
Among Banach spaces approximate injectivity is more important than injectivity. We will treat it from the point of view of enriched category theory - as enriched injectivity over complete metric spaces.
An online lecture course by the University of Münster in L-theory of rings.
We will introduce Witt groups and various flavours of L-groups and discuss some examples. We will then discuss a process called algebraic surgery. This process permits, under suitable assumptions, to simplify representatives in L-groups, and we will touch on two flavours (surgery from below and surgery from above). We will indicate how these can be used to show that various comparison maps between different L-theories are isomorphisms (in suitable ranges). Then we will go on and discuss three methods that allow for more calculations: Localisation sequences, a dévissage theorem, and an arithmetic fracture square. Using those, we will calculate the L-groups of Dedekind rings whose fraction field is a global field.
An online lecture course by the University of Münster in K-theory of forms.
In this lecture series we will describe an approach to hermitian K-theory which sheds some new light on classical Grothendieck-Witt groups of rings, especially in the domain where 2 is not assumed to be invertible. Our setup is higher categorical in nature, and is based on the concept of a Poincaré ∞-category, first suggested by Lurie. We will explain how classical examples of interest can be encoded in this setup, and how to define the principal invariants of interest, consisting of the Grothendieck-Witt spectrum and L-theory spectrum, within it. We will then describe our main abstract results, including additivity, localization and universality statements for these invariants and their relation to each other and to algebraic K-theory via the fundamental fibre sequence.
We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bi-functor with respect to the bicategorical model structure of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings.
The classical Pontryagin-Thom isomorphism equates manifold bordism groups with corresponding stable homotopy groups. This construction moreover generalizes to the equivariant context. I will discuss work which establishes a Pontryagin-Thom isomorphism for orbispaces (an orbispace is a 'space' that is locally modelled on Y/G for Y a space and G a finite group; examples of orbispaces include orbifolds and moduli spaces of pseudo-holomorphic curves). This involves defining a category of orbispectra and an involution of this category extending Spanier-Whitehead duality. Global homotopy theory also plays a key role.
In this talk we will study the lax orthogonal factorization systems (LOFSs) of Clementino and Franco, with a particular focus on finding equivalent definitions of them.
In particular, we wish to define them as a pair of classes ℰ and ℳ subject to some conditions. To achieve this, we will reduce the definition of a LOFS in terms of algebraic weak factorization systems (defined as a KZ 2-comonad L and KZ 2-monad R on the 2-category of arrows [2, 𝒞] with a 2-distributive law LR ⇒ RL) to a more property-like definition (meaning a definition with less data but more conditions).
To do this, we replace strict KZ 2-monads with the property-like definition of KZ pseudomonads in terms of Kan-extensions due to Marmolejo and Wood. In addition, pseudo-distributive laws involving KZ pseudomonads have a property-like description which will be used. Thus one can deduce the conditions the classes ℰ and ℳ must satisfy.
We will also consider some similarities and differences between LOFSs and (pseudo-)orthogonal factorization systems, and will extend their definitions to include universal fillers for squares which only commute up to a comparison 2-cell.
We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory.
The underlying building structure allows explicit computation of the K-theory. We will also present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomophic groups.
An elementary homomorphism from a free group to the pure braid group yields interesting connections between braid groups, homotopy theory, and low dimensional topology. This map induces a map on the Lie algebra obtained from the descending central series. Further, this map induces a morphism of simplicial groups. All of these maps are shown to be injective.
Brunnian braids are discussed. The analogous maps of Lie algebras induced on the filtration quotients of the mod-p descending central series is again an injection. Using these facts it turns out that the homotopy groups of this simplicial group, those of the 2-sphere, are isomorphic to natural subquotients of the pure braid group. In addition, the mod-p analogues give a connection between the classical unstable Adams spectral sequence, and the mod-p analogues of Vassiliev invariants of pure braids.
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