Related papers: Algebraic (geometric) $n$-stacks

In arXiv:2407.11958, a moduli stack parametrizing $I$--indexed diagrams of Higgs bundles over a base stack $X$ was constructed for any finite simplicial set $I$, inspiring speculations about extending the non-Abelian Hodge correspondence to…

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in…

An $(n_k)$-configuration is a set of $n$ points and $n$ lines in the projective plane such that their point-line incidence graph is $k$-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are…

We extend the stable motivic homotopy category of Voevodsky to the class of scalloped algebraic stacks, and show that it admits the formalism of Grothendieck's six operations. Objects in this category represent generalized cohomology…

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves…

Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…

We define a new algebraic structure called a \emph{pointed rack} and use it to construct ambient isotopy invariants of $ n $-braids. We first introduce an integer-valued invariant of braids using pointed racks. This is then strengthened by…

Given an algebraic stack, we compare its Nori fundamental group with that of its coarse moduli space. We also study conditions under which the stack can be uniformized by an algebraic space.

First, we prove an algebraization result for rig-smooth algebras over a general noetherian ring; this positively answers the question raised in [Sta24, Tag 0GAX]. Then we prove a general partial algebraization result in non-archimedean…

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…

We prove that cohomology and base change holds for algebraic stacks, generalizing work of Brochard in the tame case. We also show that Hom-spaces on algebraic stacks are represented by abelian cones, generalizing results of Grothendieck,…

A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…

We interpret augmented racks as a certain kind of multiplicative graphs and show that this point of view is natural for defining rack homology. We also define the analogue of the group algebra for these objects; in particular, we see how…

It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric…

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a…

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a ``cohomology theory" for complex algebraic varieties. The theory is bigraded, functorial,…

We study multisymplectic structures taking values in vector bundles with connections from the viewpoint of the Hamiltonian symmetry. We introduce the notion of bundle-valued $n$-plectic structures and exhibit some properties of them. In…

The n-string braid group of a graph X is defined as the fundamental group of the n-point configuration space of the space X. This configuration space is a finite dimensional aspherical space. A. Abrams and R. Ghrist have conjectured that…

The geometric Langlands program is distinguished in assigning spectral decompositions to all representations, not only the irreducible ones. However, it is not even clear what is meant by a spectral decomposition when one works with…

Given an abelian category and a stability condition satisfying appropriate conditions, we define generalized $K$-theoretic invariants and prove that they satisfy wall-crossing formulas. For this, we introduce a new associative algebra…