Related papers: Tannakian categories, linear differential algebrai…
We introduce the notion of a neutral representation of a finite group, or finite group scheme, $G$; a representation $V$ with the property that if a gerbe $\mathcal{G}$ over a field $k$ that is a form of the classifying stack $\mathcal{B}…
We define the Tannakian radical of a braided fusion category $\mathcal{C}$ as the intersection of its maximal Tannakian subcategories. The localization of $\mathcal{C}$ corresponding to the Tannakian radical, termed the mantle of…
We define generalized bialgebras and Hopf algebras and on this basis we introduce quantum categories and quantum groupoids. The quantization of the category of linear (super)spaces is constructed. We establish a criterion for the classical…
Let $K/F$ be a quadratic extension of $p$-adic fields, and $n$ a positive integer. A smooth irreducible representation of the group $GL(n,K)$ is said to be distinguished, if it admits on its space a nonzero $GL(n,F)$-invariant linear form.…
A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets…
We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give…
We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the…
We set up a fibred categorical theory of obstruction and classification of morphisms that specializes to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further…
We extend Goodwillie's classification of finitary linear functors to arbitrary small functors. That is we show that every small linear simplicial functor from spectra to simplicial sets is weakly equivalent to a filtered colimit of…
We present a type theory dealing with non-linear, "ordinary" dependent types (which we will call cartesian) and linear types, where both constructs may depend on terms of the former. In the interplay between these, we find new type formers…
A natural question in the theory of Tannakian categories is: What if you don't remember $\Forget$? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the \'etale…
In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and…
Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…
We investigate fibrancy conditions in the Thomason model structure on the category of small categories. In particular, we show that the category of weak equivalences of a partial model category is fibrant. Furthermore, we describe…
We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions,…
We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a…
We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial…
A tangent category is a category with an endofunctor, called the tangent bundle functor, which is equipped with various natural transformations that capture essential properties of the classical tangent bundle of smooth manifolds. In this…
A tensor extriangulated category is an extriangulated category with a symmetric monoidal structure that is compatible with the extriangulated structure. To this end we define a notion of a biextriangulated functor $\mathcal{A} \times…
Making use of Freyd's free abelian category on a preadditive category we show that if $T:D\rightarrow \mathcal{A}$ is a representation of a quiver $D$ in an abelian category $\mathcal{A}$ then there is an abelian category $\mathcal{A} (T)$,…