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.
Tag - Category theory
An LMS online lecture course in differential categories.
Cartmell showed that the category of generalized algebraic theories is equivalent to the category of contextual categories. This implies that the theory of generalized algebraic theories is essentially algebraic. We characterize the essentially algebraic theory of generalized algebraic theories as the free category with finite limits and with an exponentiable arrow.
Hopf categories were introduced by Batista, Caenepeel and Vercruysse in 2016, as a many-object generalization of Hopf algebras linked to other notions like multiplier Hopf algebras and with applications to categorical Galois theory. What is of particular interest is that the multiplication and comultiplication appear to make use of different monoidal products: Gabriella Böhm in subsequent work expressed Hopf categories as specific opmonoidal monads. In our work, we follow a different direction of generalizing Hopf monoids in a braided monoidal bicategory, that allows us to realize Hopf categories as Hopf-type objects over the same monoidal product, restoring in a sense the self-dual feature of classical Hopf algebras. In this talk, we introduce oplax bimonoids and oplax Hopf monoids in an arbitrary braided monoidal bicategory, we study their main properties and we exemplify such structures in a Span-type bicategory where they return semi-Hopf and Hopf categories.
Comprehension schemes arose as crucial notions in the early work on the foundations of set theory, and hence they found expression in a considerable variety of foundational settings for mathematics. Particularly, they have been introduced to the context of categorical logic first by Lawvere and then by Benabou in the 1970s.
In this talk we define and study a theory of comprehension schemes for fibered ∞-categories, generalizing Johnstone’s respective notion for ordinary categories. This includes natural generalizations of all the fundamental instances originally defined by Benabou, and their application to Jacob's comprehension categories. Thereby, we can characterize
- numerous categorical structures arising in higher topos theory,
- the notion of univalence,
- internal ∞-categories,
in terms of comprehension schemes, while some of the 1-categorical counterparts fail to hold in ordinary category theory. As an application, we can show that the universal cartesian fibration is represented via externalization by the "freely walking chain" in the ∞-category of small ∞-categories.
In the end, if my time management permits, we take a look at the externalization construction of internal ∞-categories from a model categorical perspective and review some examples from the literature in this light.
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.
In this talk I will introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application of this duality, it follows that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. I will also comment on a conceptual approach to this problem that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG-algebras and DG-bimodules.
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.
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