Lauren Bandklayder, Julia E. Bergner, Rhiannon Griffiths, Brenda Johnson, Rekha Santhanam
In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.
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Replacing functors with enriched ones
本研究针对（带点）单纯集等多种强化基，提出给定函子自然等价于强化函子的简单准则，并给出了Lurie关于Dwyer-Kan局部化结果的新证明等推论。
Cofibrantly generated model structures for functor calculus
本文针对函子演算，给出了判定通过 Bousfield 定理获得的模型范畴何时为余纤维生成的一般性准则，并以此恢复和包含了多种已有的同伦函子与离散函子演算模型结构。
Homotopy Theory of Enriched Mackey Functors
本文系统发展了闭多范畴上富集的图表、预层与 Mackey 函子理论，研究富集改变及其复合性，并将其应用于由 K-理论多函子诱导的同伦理论等价，涵盖带点多范畴、置换范畴和对称谱等情形。
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 functors, and show that this category, equipped with the directional derivative, is a cartesian differential category in the sense of Blute, Cockett, and Seely. This provides an abstract framework that makes certain analogies between classical and functor calculus explicit. Inspired by Huang, Marcantognini, and Young's chain rule for higher order directional d…
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 then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct…
One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when $n = 1$. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al. This pap…
Functor calculus via non-cubes
本文将Goodwillie函子微积分推广到非立方体索引图，构造了更广图类的万有切除逼近并扩展Taylor塔，同时证明其极限与原Taylor塔一致，并探讨了新切除概念与经典概念的重合条件。
We introduce a notion of bimodule in the setting of enriched $\infty$-categories, and use this to construct a double $\infty$-category of enriched $\infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natural definition of natural transformations in this context, and show that in the underlying $(\infty,2)$-category of enriched $\infty$-categories with functors as 1-morphisms the 2-morphisms are given by natural transformations.
On bi-enriched $\infty$-categories
本文将Lurie的富集$\infty$-范畴推广为左富集、右富集和双富集$\infty$-范畴，并利用双富集结构研究富集函子范畴、富集Kan扩张及相关end/coend公式，发展了独立的富集$\infty$-范畴理论。
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 allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techni…
Functor calculus completions for retractive operadic algebras in spectra
本文在Goodwillie同伦函子演算框架下研究谱中回缩operad代数的Bousfield-Kan完备化及恒等函子Taylor塔的收敛性。
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 layer of which is classified by a spectrum with an action of $C_2 \ltimes U(n)$. We further give model categorical considerations, producing a zig-zag of Quillen equivalences between spectra with an action of $C_2 \ltimes U(n)$ and a model structure on the category of input functors which captures the homotopy theory of the $n$-th layer of the Taylor tower.
We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally-enriched functor from a spectrally-enriched version of permutative categories to the category of spectra that is built using an appropriate version of K-theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg--MacLane spectra for Mackey functors and for…
We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly-generated infinity-categories. The first theorem say that the construction of higher derivatives for spectrum-valued functors is a Day convolution of copies of the first derivative construction. The second theorem says that the derivatives of any functor can be realized as natural transformation objects for derivatives of spectrum-valued functors. Together these results allow us to construct an infinity-operad that models the derivatives…
Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce "multivariable" manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. We construct multivariable Taylor approximations to such functors, classify multivariable homogeneous functors, apply this classification to compute the derivatives of a functor, and show what this gives for the space of link maps. We also relate Taylor approximations in single v…
A Context for Manifold Calculus
本文在∞-范畴中发展了Weiss的流形微分算术，探讨多项式函子、Kan扩展与Weiss sheaves之间的联系，并分类同质函子，推广Weiss和Boavida de Brito的定理。
A Dwyer-Rezk classification for polynomial functors in Weiss calculus
本文通过不同方法证明了Weiss微分学中多项式函子的分类，并给出了同质函子分类的新证明。
For a smooth manifold M, we define a topological space X(k,M), and show that polynomial functors O(M)--> C of degree <= k from the poset of open subsets of M to a simplicial model category can be classified be a version of linear functors from O(X(k,M)) to C.
Naturality of the $\infty$-Categorical Enriched Yoneda Embedding
本文通过将 Hinich 的 ∞-范畴丰富 Yoneda 嵌入构造为丰富预层函子与 Heine 函子之间部分伴随的单位，使其自然化，并进一步证明该部分伴随在原子对象上限制为真正的伴随。
Recent work of Biedermann and Röndigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with an action of the…
Recovering unitary calculus from calculus with reality
本文证明酉函子微积分可从带实性的酉函子微积分中完全恢复，类比于复拓扑$K$-理论通过遗忘$C_2$-作用从带实性的$K$-理论中恢复。
Parametrised functor calculus: excision, spheres, and semiadditivity
本研究建立参数化函子微积分基础，引入可切除偏序集与可切除逼近理论，通过原子轨道范畴上两类可切除偏序集的对比，关联特定球面可逆性与参数化半加性，推广等变同伦论经典结果。
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 construction of polynomial functors, the classification of homogeneous functors, and results regarding convergence of the Taylor tower. We sprinkle examples throughout, and pay special attention to spaces of smooth embeddings.
We show that the category of $n$-excisive functors from the $\infty$-category of spectra to a target stable $\infty$-category $\mathbf{E}$ is equivalent to the category of $\mathbf{E}$-valued Mackey functors on an indexing category built from finite sets and surjections. This new classification of polynomial functors arises from an investigation of the structure present on cross effects. The path to this result involves a pair of surprising extension theorems for polynomial functors and a discussion of some interesting topics in semiadditive $\infty$- category theory, including a formula for t…
Classification of homogeneous functors in manifold calculus
本文在单纯模型范畴中构造拓扑空间Â，对在k个开球上取值等价于给定对象A的齐性函子进行分类，推广了Weiss关于拓扑空间值齐性函子的分类结果。
The discriminant method is a tool for describing the cohomology, or the homotopy type, of certain spaces of smooth maps with uncomplicated singularities from a smooth compact manifold L to R^k. We recast some of it in the language of functor calculus. This reformulation allows us to use the discriminant method in a setting where we wish to impose conditions on the multilocal behavior of smooth maps f from L to R^k.
It is known that there is a weak-equivalence between the geometric realization of a simplicially enriched small category and its cofibrant replacement [12]. In this paper, we show that when only small categories are considered there exists a homeomorphism between these geometric realizations. We also discuss the naturality of these homoemorphisms. The inclusion of the category of small categories to the category of simplicially enriched categories, the cofibrant replacement of simplicially enriched categories, and the geometric realization of simplicially enriched categories are three composab…
This is a continuation, completion, and generalization of our previous joint work with B. Chorny. We supply model structures and Quillen equivalences underlying Goodwillie's constructions on the homotopy level for functors between simplicial model categories satisfying mild hypotheses.
We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. Duoidal categories (introduced by Aguillar and Mahajan under the name 2-monoidal categories) are categories with two monoidal structures which are related by some, not necessary invertible, coherence morphisms. Centers of monoids in this sense include many examples which are not `classical.' In particular, the 2-category of categories is an example of a center in our sense. Examples of homotopy center (analogue of the classical Hochschild complex) include …
We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, these data determine a bimonoidal functor. We extend this result to $n$-variables, and prove that, in a manner analogous to that of butterflies, these multi-extensions can be composed. This is phrased in terms of a multilinear functor calculus in a bicategory. As an application, we study a bimonoidal ca…
We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite type knot invariants, etc. Some open questions and conjectures of interest are given throughout.
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 discrete calculus tower for functors from an unbased simplicial model category to chain complexes over a fixed commutative ring. Much of the construction of the Taylor tower for functors to spectra carries over to this context. However, one of the essential steps in the construction requires proving that a particular functor is part of a cotriple. For this, one ne…
We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Ka…
We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n-1)-excisive part, can be classified: they correspond to symmetric functors of n variables that are reduced and 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen's algebraic K-theory.
Cross effects for functors from posets
本文研究了格路上泛函的泛函微分学与多参数持久模块的投射维数关系，提出了新的泛函微分学概念，并证明了多参数持久模块投射维数的充要条件。
The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the $n$-homogeneous model structure, the $n$-th derivative is a Quillen functor to the category of spectra with $Σ_n$-action. After taking into account only finitary functors -- which …
It is known that in the integral cohomology of BSO(2m), the square of the Euler class is the same as the Pontryagin class in degree 4m. Given that the Pontryagin classes extend rationally to the cohomology of BSTOP(2m), it is reasonable to ask whether the same relation between the Euler class and the Pontryagin class in degree 4m is still valid in the rational cohomology of BSTOP(2m). In this paper we use smoothing theory and tools from homotopy theory to reformulate the hypothesis, and variants, in a differential topology setting and in a functor calculus setting.
We associate a Taylor tower supplied by calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral sequence collapses along this line and that the Taylor tower represents a universal finite type knot invariant.
We define the notion of an enriched Reedy category, and show that if A is a C-Reedy category for some symmetric monoidal model category C and M is a C-model category, the category of C-functors and C-natural transformations from A to M is again a model category.
This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie's homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.
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, there is a Weiss tower associated to the functor which sends a vector space V to F evaluated at the one-point compactification of V. In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Ta…
Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the $h$-principle, we show that for a symplectic manifold $N$, the analytic approximation to the Lagrangian embeddings functor $\mathrm{Emb}_{\mathrm{Lag}}(-,N)$ is the totally real embeddings functor $\mathrm{Emb}_{\mathrm{TR}}(-,N)$. More generally, for subsets $\mathcal{A}$ of the $m$-plane Grassmannian bundle $\mathrm{Gr}(m,TN)$ for which the $h$-principle holds for $\…
We study a connection between mapping spaces of bimodules and of infinitesimal bimodules over an operad. As main application and motivation of our work, we produce an explicit delooping of the manifold calculus tower associated to the space of smooth maps $D^{m}\rightarrow D^{n}$ of discs, $n\geq m$, avoiding any given multisingularity and coinciding with the standard inclusion near $\partial D^{m}$. In particular, we give a new proof of the delooping of the space of disc embeddings in terms of little discs operads maps with the advantage that it can be applied to more general mapping spaces.
Embedding Calculus, as described by Weiss, is a calculus of functors, suitable for studying contravariant functors from the poset of open subsets of a smooth manifold M, denoted O(M), to a category of topological spaces (of which the functor Emb(-,N) for some fixed manifold N is a prime example). Polynomial functors of degree k can be characterized by their restriction to O_k(M), the full subposet of O(M) consisting of open sets which are a disjoint union of at most k components, each diffeomorphic to the open unit ball. In this work, we replace O_k(M) by more general subposets and see that we…
On the tensor product of enriched $\infty$-categories
本文证明充实∞-范畴的张量积保持余极限并修正了前人错误，同时证明本质满射与完全忠实函子构成分解系统且与张量积及内部Hom兼容。
We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial G-sets to symmetric G-spectra, where G is a finite group. We extend a notion of G-linearity suggested by Blumberg to define stably excisive and rho-analytic homotopy functors, as well as a G-differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected G-maps to G-equivalences. It is analogous to the classical result of Goodwillie that "functors with zero derivative are locally constant". As main example we show that Hes…
We show that one can use model categories to construct rational orthogonal calculus. That is, given a continuous functor from vector spaces to based spaces one can construct a tower of approximations to this functor depending only on the rational homology type of the input functor, whose layers are given by rational spectra with an action of $O(n)$. By work of Greenlees and Shipley, we see that these layers are classified by torsion $H^*(B SO(n))[O(n)/SO(n)]$-modules.
Dwyer-Kan homotopy theory of enriched categories
本文在满足幺半群公理的组合闭对称幺半模范畴上的富集小范畴范畴中，建立了以 Dwyer-Kan 等价为弱等价的模型结构。
This is a (slightly edited) version of the PhD dissertation of the author, submitted to Brown University in July 2005. We construct a homotopy calculus of functors in the sense of Goodwillie for the categories of rational homotopy theory. More precisely, given a homotopy functor between any of the categories of differential graded vector spaces (DG), reduced differential graded vector spaces, differential graded Lie algebras (DGL), and differential graded coalgebras (DGC), we show that there is an associated approximating "rational Taylor tower" of excisive functors. The fibers in this tower a…
A homotopy coherent nerve for $(\infty,n)$-categories
本文为 $(\infty,n)$-范畴构造同伦相干神经，证明其给出严格丰富范畴与 Segal 范畴对象模型间的右 Quillen 等价，从而统一两种同伦相干图定义。
We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum dF(X) of such a functor F at a simplicial set X can be equipped with a right action by the loop group of its domain X, and a free left action by the loop group of its codomain Y = F(X). The derivative spectrum d(E o F)(X)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives dE(Y) and dF(X), with respect to the two actions of the loop group of Y. As an …
Enriched $\infty$-categories as marked module categories
本研究证明富化$\u221e$-范畴可由带可表预层标记的富化预层范畴完全刻画，建立了其与带$\u221e$-群胚函子的可展示$\u2113$-模范畴满子范畴的等价，借此构造新的富化$\u221e$-范畴张量积并重新表述单值性概念。
Given a compact manifold X, the set of simple manifold structures on X x Δ^k relative to the boundary can be viewed as the k-th homotopy group of a space §^s (X). This space is called the block structure space of X. We study the block structure spaces of real projective spaces. Generalizing Wall's join construction we show that there is a functor from the category of finite dimensional real vector spaces with inner product to the category of pointed spaces which sends the vector space V to the block structure space of the projective space of V. We study this functor from the point of view of o…
The Taylor tower of a functor from based spaces to spectra can be classified according to the action of a certain comonad on the collection of derivatives of the functor. We describe various equivalent conditions under which this action can be lifted to the structure of a module over the Koszul dual of the little L-discs operad. In particular, we show that this is the case when the functor is a left Kan extension from a certain category of `pointed framed L-manifolds' and pointed framed embeddings. As an application we prove that the Taylor tower of Waldhausen's algebraic K-theory of spaces fu…
Lax monoidal adjunctions, two-variable fibrations and the calculus of mates
本文建立面向 ∞-范畴函子的配偶演算，推广 unstraightening 等价，并通过二变量纤维化刻画 lax/oplax 幺半伴随及 Gray 张量积函子。
A simple characterization of Quillen adjunctions
本文研究了模型范畴间的富化右伴随函子，指出保持无环纤维化和纤维对象的函子通常即为右Quillen函子，提供了Quillen伴随的简单刻画。
Lax monoidality for products of enriched higher categories
本文证明松弛 $\mathbb{E}_{n+1}$-幺半函子可诱导富集 $\infty$-范畴间的松弛 $\mathbb{E}_n$-幺半函子，并基于 Boardman--Vogt 张量积给出将图示构造推广至 $\mathcal{O}$-幺半范畴的准则。
Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Let C be a category that has a zero object and all small limits. A homogeneous functor (in the sense of manifold calculus) of degree k from O(M) to C is called very good if it sends isotopy equivalences to isomorphisms. In this paper we show that the category VGHF of such functors is equivalent to the category of contravariant functors from the fundamental groupoid of Conf(k, M) to C, where Conf(k, M) stands for the unordered configuration space of k points in M. As a consequence of this result, we show that the catego…
Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice the embedding dimension of M, the Taylor tower in the sense of orthogonal calculus (henceforward called ``the orthogonal tower'') of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence…
A Categorical Approach to Möbius Inversion via Derived Functors
本文通过丰富范畴中的导出函子定义Möbius上同调，证明其欧拉示性数恢复经典Möbius反演，并给出Rota的Galois联络的范畴版本，统一了组合学与同调代数。
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