Related papers: Algebraicity in monochromatic homotopy theory

We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B…

Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $\Bbbk$ of characteristic $p\ge 0$. The category of locally finite dimensional representations of $A $ is…

We develop a notion of covariant differential calculus for Hopf algebroids. As a byproduct, we prove analogues of the fundamental theorem of Hopf modules and a Takeuchi-Schneider equivalence in the realm of Hopf algebroids. The resulting…

We introduce a natural generalization of the definition of a symmetric Hopf algebroid, internal to any symmetric monoidal category with coequalizers that commute with the monoidal product. Motivation for this is the study of Heisenberg…

We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\mathbb{P}^\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of…

In this paper, we describe a novel way of identifying Adams spectral sequence $E_2$-terms in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on…

In this paper we prove an $\infty$-categorical version of the reflection theorem of Ad\'amek-Rosick\'y. Namely, that a full subcategory of a presentable $\infty$-category which is closed under limits and $\kappa$-filtered colimits is a…

Algebraic $kk$-theory, introduced by Corti\~nas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor…

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…

We provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role…

We prove that both the local and global $\mathbb{A}^1$-degree of an endomorphism of affine space can be computed in terms of the multivariate B\'ezoutian. In particular, we show that the B\'ezoutian bilinear form, the Scheja--Storch form,…

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to…

The well-known theorem of Eilenberg and Ganea expresses the Lusternik - Schnirelmann category of an aspherical space as the cohomological dimension of its fundamental group. In this paper we study a similar problem of determining…

There are at least two ways to approach the homotopy theory of spaces `at chromatic height $n$': one may localize with respect to $T(n)$-homology or with respect to $v_n$-periodic homotopy groups. It was already observed by Bousfield that…

We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion…

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes…

We construct a variant $\mathcal{K}_n$ of the Hopf algebra $\mathcal{H}_n$, which acts directly on the noncommutative model for the generic space of leaves rather than on its frame bundle. We prove that the Hopf cyclic cohomology of…

Fix an odd prime $p$. Let $X$ be a pointed space whose $p$-completed K-theory $\mathrm{KU}_p^*(X)$ is an exterior algebra on a finite number of odd generators; examples include odd spheres and many H-spaces. We give a…

We introduce and study the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups. We realize some results of chromatic homotopy theory in terms of the geometry of this stack. For…

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of ``additive groupoid enriched categories'', in which a secondary analog of homological algebra can…