Related papers: Strict \infty-groupoids are Grothendieck \infty-gr…

We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category.…

We define a functor which takes in an $(\infty,1)$-category and outputs an $(\omega,1)$-category, the natural maximally "strict" version of an $(\infty,1)$-category. We do this by modeling $(\infty,1)$-categories as categories enriched in…

We prove that given any strong, stable derivator and a $t$-structure on its base triangulated category $\cal D$, the $t$-structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely…

We prove that the torsion-free lamplighter group $\Gamma = \mathbb{Z}^n \wr \mathbb{Z}$ of any rank $n \in \mathbb{N}$ is profinitely rigid in the absolute sense: the finite quotients of $\Gamma$ determine its isomorphism type uniquely…

We demonstrate that any full and faithful $*$-functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such…

We prove that the action of a generalized braid group on an enhanced triangulated categories, generated by spherical twist functors along an ADE-configuration of $\omega$-spherical objects, is faithful for any integer $\omega \neq 1$.

Let A be a commutative noetherian ring. Call a functor <<commutative A-algebras>> --> <<sets>> coherent if it can be built up (via iterated finite limits) from functors of the form B \mapsto M tensor_A B, where M is a f.g. A-module. When…

We show that in $K$-theory-like categories many corner embeddings into a discrete algebra of compact operators are invertible, and consequently functors on splitexact algebraic $KK$-theory are faithful if and only if they are faithful on…

In this short note, we classify linear categorified open topological field theories in dimension two by pivotal Grothendieck-Verdier categories, a type of monoidal category equipped with a weak, not necessarily rigid duality. In combination…

Let X be a smooth variety over a field of characteristic p>0. We prove that the forgetful functor from the category of overconvergent F-isocrystals on X to the category of convergent F-isocrystals is fully faithful. The argument uses the…

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-injective objects to be a torsion class. By doing so, we propose a notion of $n$-hereditary categories. We also define and…

We show that for any type in Martin-L\"of Intensional Type Theory, the terms of that type and its higher identity types form a weak omega-category in the sense of Leinster. Precisely, we construct a contractible globular operad of definable…

We study weakly invertible cells in weak $\omega$-categories in the sense of Batanin-Leinster, adopting the coinductive definition of weak invertibility. We show that weakly invertible cells in a weak $\omega$-category are closed under…

We show that for any separably closed field $k$ of characteristic $p>0$, the canonical functor from nilpotent $p$-adic spaces to $\mathbb{E}_{\infty}$-coalgebras over $k$ (given by singular chains with coefficients in $k$) is fully…

We study stability properties of fully faithful functors, and compute mapping anima in pushouts of $\infty$-categories along fully faithful functors. We provide applications of these calculations to pushouts along Dwyer functors and Reedy…

In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $\Theta^{\infty}_{\mathbb{M}^m}$ whose set-models are…

In this paper, we prove that all finitely generated 3-manifold groups are Grothendieck rigid. More precisely, for any finitely generated 3-manifold group $G$ and any finitely generated proper subgroup $H<G$, we prove that the inclusion…

It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a…

We show how to treat families of $\infty$-categories fibered in categorical patterns (e.g., $\infty$-operads and monoidal $\infty$-categories) in terms of fibrations by relativizing the Grothendieck construction. As applications, we…

We study the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor. In particular, we prove the categorical entropy of the twist functor coincides with that of the cotwist functor if the…