Corpus ID: 48943025


  author={Aaron Mazel-Gee and N. Rozenblyum},
For an arbitrary symmetric monoidal∞-category V, we define the factorization homology of V-enriched∞-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace map. As we show in [AMGRa], these induce their stable counterparts through linearization (in the sense… Expand
Traces for factorization homology in dimension 1
We construct a circle-invariant trace from the factorization homology of the circle trace : ∫ α S1 End(V ) −→ 1 associated to a dualizable object V ∈ X in a symmetric monoidal ∞-category. This provesExpand
The space of traces in symmetric monoidal infinity categories
In any symmetric monoidal 1-category the trace assigns to an endomorphism of a dualisable object an endomorphisms of the unit object. The trace is natural with respect to symmetric monoidal functorsExpand
Equivariant localization and completion in cyclic homology and derived loop spaces
We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces asExpand
A naive approach to genuine $G$-spectra and cyclotomic spectra
For any compact Lie group $G$, we give a description of genuine $G$-spectra in terms of the naive equivariant spectra underlying their geometric fixedpoints. We use this to give an analogousExpand
Lectures on Factorization Homology, ∞-Categories, and Topological Field Theories
These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are notExpand
Shadows are Bicategorical Traces
The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. UsingExpand
On curves in K-theory and TR
We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line S[t] as a functor defined on the∞-category of cyclotomic spectra with values inExpand


A cartesian presentation of weak n–categories
We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category ofExpand
Enriched ∞-categories via non-symmetric ∞-operads
Abstract We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞-category. Our theory of enriched ∞-categories has many desirable properties; for instance, ifExpand
The smooth Whitehead spectrum of a point at odd regular primes
Let p be an odd regular prime, and assume that the Lichtenbaum{Quillen conjecture holds for K(Z[1=p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh () is described. AExpand
The cyclotomic trace and algebraic K-theory of spaces
The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, andExpand
Power maps and epicyclic spaces
Abstract In this paper, we present the basic facts of the theory of epicyclic spaces for the first time considered by T. Goodwillie in an unpublished letter to F. Waldhausen. An epicyclic space is aExpand
Cyclic polytopes and the $K$-theory of truncated polynomial algebras
This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x isExpand
On the K-theory of truncated polynomial algebras over the integers
We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopyExpand
Waldhausen’s algebraic K-theory of spaces is a homotopy functor X 7→ A(X) from spaces to infinite loop spaces. It is related to diffeomorphisms of smooth manifolds through stable smoothExpand
Crossed simplicial groups and their associated homology
We introduce a notion of crossed simplicial group, which generalizes Connes' notion of the cyclic category. We show that this concept has several equivalent descriptions and give a completeExpand
The Geometry of Iterated Loop Spaces
i Preface This it the first of a series of papers devoted to the study of iterated loop spaces. Our goal is to develop a simple coherent theory which encompasses most of the known results about suchExpand