相关论文: Manifolds, K-theory and the calculus of functors

Let F be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of F have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little…

We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which…

We introduce a version of Koszul duality for categories, which extends the Koszul duality of operads and right modules. We demonstrate that the derivatives which appear in Weiss calculus (with values in spectra) form a right module over the…

We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study "functors with reality" such as the Real classifying space functor, $BU_\mathbb{R}(-)$. The calculus produces a Taylor tower, the $n$-th…

We present an introduction to the manifold calculus of functors, due to Goodwillie and Weiss. Our perspective focuses on the role the derivatives of a functor F play in this theory, and the analogies with ordinary calculus. We survey the…

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the…

We describe Taylor towers for spaces of knots arising from Goodwillie-Weiss calculus of the embedding functor and extend the configuration space integrals of Bott and Taubes from spaces of knots to the stages of the towers. We show that…

We show that Koszul duality for operads in $(\mathrm{Top},\times)$ can be expressed via generalized Thom complexes. As an application, we prove the Koszul self duality of the little disk modules $E_M$. We discuss implications for…

A functor on compact Hausdorf spaces is constructed as the sum of certain equivariant K-theory groups. It is shown that the functor takes values in lambda-rings and satisfies a Thom isomorphism. In the case that the space is a CW-complex…

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…

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of $\mathbb{R}^m$ into $\mathbb{R}^n$. We view the space of embeddings as the value of a certain functor at…

Let R be a discrete unital ring, and let M be an R-bimodule. We extend Waldhausen's equivalence from the suspension of the Nil K-theory of R with coefficients in M to the K theory of the tensor algebra T_R(M), and get a map from the…

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose…

We show that the Kontsevich operad, as an operad with multiplication, provides a model for the Taylor tower of the functor defined by taking the homotopy fiber of the inclusion of embeddings of an interval in a cube to the corresponding…

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular,…

We develop a generalization of manifold calculus in the sense of Goodwillie-Weiss where the manifold is replaced by a simplicial complex. We consider functors from the category of open subsets of a fixed simplical complex into the category…

Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a…

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria…

Manifold calculus is a form of functor calculus concerned with functors from some category of manifolds to spaces. A weakness in the original formulation is that it is not continuous in the sense that it does not handle well the natural…

We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We…