Related papers: Cross effects and calculus in an unbased setting
We reconceptualize the process of forming $n$-excisive approximations to $\infty$-categories, in the sense of Heuts, as inverting the suspension functor lifted to $A_n$-cogroup objects. We characterize $n$-excisive $\infty$-categories as…
The objective of this paper is to provide a fairly general model category context in which one can perform Goodwille Calculus. There are two main parts to the paper, the first establishing general conditions which guarantee the existence of…
We define a bar construction endofunctor on the category of commutative augmented monoids $A$ of a symmetric monoidal category $\mathcal{V}$ endowed with a left adjoint monoidal functor $F:s\mathbf{Set}\to \mathcal{V}$. To do this, we need…
We introduce general methods to analyse the Goodwillie tower of the identity functor on a wedge $X \vee Y$ of spaces (using the Hilton-Milnor theorem) and on the cofibre $\mathrm{cof}(f)$ of a map $f: X \rightarrow Y$. We deduce some…
Depth three and finite depth are notions known for subfactors via diagrams and Frobenius extensions of rings via centralizers in endomorphism towers. From the point of view of depth two ring extensions, we provide a clear definition of…
We call a tensor functor $F:\mathcal{C}\to\mathcal{D}$ between finite tensor categories $\otimes$-Frobenius if its left and right adjoints are isomorphic as $\mathcal{C}$-bimodule functors. We give several characterizations of this notion…
In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully construct a category of abelian categories and suitably homotopically defined…
In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a…
In their article "Elementary construction of perverse sheaves", R.MacPherson and K. Vilonen show that on a Thom-Mather space X the category PervX of perverse sheaves is equivalent to the category C(F, G, T) whose objects are data of…
We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which…
Generalizing work by Pinzari and Roberts, we characterize actions of a compact quantum group G on C*-algebras in terms of what we call weak unitary tensor functors from Rep G into categories of C*-correspondences. We discuss the relation of…
We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we…
In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension…
In earlier work, we introduced the `Monster tower', a tower of fibrations associated to planar curves. We constructed an algorithm for classifying its points with respect to the equivalence relation generated by the action of the contact…
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of…
We study several structure aspects of functor categories from a small additive category to a module category, in particular the category F(A,K) of functors from finitely generated free modules over a commutative ring A to vector spaces over…
We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group $W$, in terms of the spectrum of an associated operator,…
Given a suitable functor T:C -> D between model categories, we define a long exact sequence relating the homotopy groups of any X in C with those of TX, and use this to describe an obstruction theory for lifting an object G in D to C.…
We show an isomorphism between the kernel of the C*-algebra of the tangent groupoid of a filtered manifold and the crossed product of the order 0 pseudodifferential operators in the associated filtered calculus by a natural R-action. This…
Consider a cofibrantly generated model category $S$, a small category $C$ and a subcategory $D$ of $C$. We endow the category $S^C$ of functors from $C$ to $S$ with a model structure, defining weak equivalences and fibrations objectwise but…