Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTHH and traces of enriched categories
94
0
0.0
(
0
)
Authors
John D. Berman
Ask ChatGPT about the research
No Arabic abstract
We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched infinity category. Our results rely crucially on an elementary, model-independent framework for enriched higher category theory, which may be of independent interest.
rate research
Read More
We prove a version of J.P. Mays theorem on the additivity of traces, in symmetric monoidal stable $infty$-categories. Our proof proceeds via a categorification, namely we use the additivity of topological Hochschild homology as an invariant of stable $infty$-categories and construct a morphism of spectra $mathrm{THH}(mathbf C)to mathrm{End}(mathbf 1_mathbf C)$ for $mathbf C$ a stably symmetric monoidal rigid $infty$-category. We also explain how to get a more general statement involving traces of finite (homotopy) colimits.
We define Grothendieck-Witt spectra in the setting of Poincare $infty$-categories and show that they fit into an extension with a L- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and generalisations of Karoubis fundamental and periodicity theorems for rings in which 2 need not be invertible. Our set-up allows for the uniform treatment of such algebraic examples alongside homotopy-theoretic generalisations: For example, the periodicity theorem holds for complex oriented $mathrm{E}_1$-rings, and we show that the Grothendieck-Witt theory of parametrised spectra recovers Weiss and Williams LA-theory. Our Grothendieck-Witt spectra are defined via a version of the hermitian Q-construction, and a novel feature of our approach is to interpret the latter as a cobordism category. This perspective also allows us to give a hermitian version -- along with a concise proof -- of the theorem of Blumberg, Gepner and Tabuada, and provides a cobordism theoretic description of the aforementioned LA-spectra.
For every $infty$-category $mathscr{C}$, there is a homotopy $n$-category $mathrm{h}_n mathscr{C}$ and a canonical functor $gamma_n colon mathscr{C} to mathrm{h}_n mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Based on the idea of the homotopy $n$-category, we introduce the notion of an $n$-derivator and study the main examples arising from $infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $infty$-derivators and prove that the canonical comparison map from the Waldhausen $K$-theory of $mathscr{C}$ to the $K$-theory of the associated $n$-derivator $mathbb{D}_{mathscr{C}}^{(n)}$ is $(n+1)$-connected. We also prove that this comparison map identifies derivator $K$-theory of $infty$-derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy $n$-category, we define also a $K$-theory space $K(mathrm{h}_n mathscr{C}, mathrm{can})$ associated to $mathrm{h}_n mathscr{C}$. We prove that the canonical comparison map from the Waldhausen $K$-theory of $mathscr{C}$ to $K(mathrm{h}_n mathscr{C}, mathrm{can})$ is $n$-connected.
This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In this article we lay the foundations of our approach by considering Luries notion of a Poincare $infty$-category, which permits an abstract counterpart of unimodular forms called Poincare objects. We analyse the special cases of hyperbolic and metabolic Poincare objects, and establish a version of Ranickis algebraic Thom construction. For derived $infty$-categories of rings, we classify all Poincare structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincare structures on $infty$-categories of parametrised spectra, recovering the visible signature of a Poincare duality space. We conduct a thorough investigation of the global structural properties of Poincare $infty$-categories, showing in particular that they form a bicomplete, closed symmetric monoidal $infty$-category. We also study the process of tensoring and cotensoring a Poincare $infty$-category over a finite simplicial complex, a construction featuring prominently in the definition of the L- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0-th Grothendieck-Witt group of a Poincare $infty$-category using generators and relations. We extract its basic properties, relating it in particular to the 0-th L- and algebraic K-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal properties. In this paper, we introduce enriched presheaves on an enriched infinity category. We prove analogues of most familiar properties of presheaves. For example, we compute limits and colimits of presheaves, prove that all presheaves are colimits of representable presheaves, and prove a version of the Yoneda lemma.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
