### Publications and preprints

*Dual tangent structures for ∞-toposes*, arXiv:2101.08805, 18 pages, submitted.We describe dual notions of tangent bundle for an ∞-topos, each underlying a tangent ∞-category in the sense of Bauer, Burke and the author. One of those notions is Lurie's tangent bundle functor for presentable ∞-categories, and the other is its adjoint. We calculate that adjoint for injective ∞-toposes, where it is given by applying Lurie's tangent bundle on ∞-categories of points.

*Tangent ∞-categories and Goodwillie calculus*, (joint with Kristine Bauer and Matthew Burke), arXiv:2101.07819, 128 pages, submitted.We make precise the analogy between Goodwillie's calculus of functors in homotopy theory and the differential calculus of smooth manifolds by introducing a higher-categorical framework of which both theories are examples. That framework is an extension to ∞-categories of the tangent categories of Cockett and Cruttwell (introduced originally by Rosický). A tangent structure on an ∞-category X consists of an endofunctor on X, which plays the role of the tangent bundle construction, together with various natural transformations that mimic structure possessed by the ordinary tangent bundles of smooth manifolds and that satisfy similar conditions.

The tangent bundle functor in Goodwillie calculus is Lurie's tangent bundle for ∞-categories, introduced to generalize the cotangent complexes of André, Quillen and Illusie. We show that Lurie's construction admits the additional structure maps and satisfies the conditions needed to form a tangent ∞-category, which we refer to as the Goodwillie tangent structure on the ∞-category of ∞-categories.

Cockett and Cruttwell (and others) have started to develop various aspects of differential geometry in the abstract context of tangent categories, and we begin to apply those ideas to Goodwillie calculus. For example, we show that the role of Euclidean spaces in the calculus of manifolds is played in Goodwillie calculus by the stable ∞-categories. We also show that Goodwillie's n-excisive functors are the direct analogues of n-jets of smooth maps between manifolds; to state that connection precisely, we develop a notion of tangent (∞,2)-category and show that Goodwillie calculus is best understood in that context.

*Koszul duality for topological E*, (joint with Paolo Salvatore), arXiv:2002.03878, 54 pages, submitted._{n}-operadsWe show that the Koszul dual of an E

_{n}-operad in spectra is O(n)-equivariantly equivalent to its n-fold desuspension. To this purpose we introduce a new O(n)-operad of Euclidean spaces R_{n}, the barycentric operad, that is fibred over simplexes and has homeomorphisms as structure maps; we also introduce its sub-operad of restricted little n-discs D_{n}, that is an E_{n}-operad. The duality is realized by an unstable explicit S-duality pairing (F_{n})_{+}∧ BD_{n}→ S_{n}, where B is the bar-cooperad construction, F_{n}is the Fulton-MacPherson E_{n}-operad, and the dualizing object S_{n}is an operad of spheres that are one-point compactifications of star-shaped neighbourhoods in R_{n}. We also identify the Koszul dual of the operad inclusion map E_{n}→ E_{n+m}as the (n+m)-fold desuspension of an unstable operad map E_{n+m}→ Σ^{m}E_{n}defined by May.*Infinity-operads and Day convolution in Goodwillie calculus*, Journal of the London Mathematical Society**104(3)**, (2021), 1204-1249.We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly-generated ∞-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 ∞-operad that models the derivatives of the identity functor on any pointed compactly-generated ∞-category.

Our main example is the ∞-category of algebras over a stable ∞-operad, in which case we show that the derivatives of the identity essentially recover the same ∞-operad, making precise a well-known slogan in Goodwillie calculus.

We also describe a bimodule structure on the derivatives of an arbitrary functor, over the ∞-operads given by the derivatives of the identity on the source and target, and we conjecture a chain rule that generalizes previous work of Arone and the author in the case of functors of pointed spaces and spectra.

*Goodwillie Calculus*, (joint with Gregory Arone), Handbook of Homotopy Theory, CRC Press (2019), 1--38.We survey the theory and applications of Goodwillie's calculus of homotopy functors and related topics.

*Derived Koszul duality and TQ-homology completion of structured ring spectra*, (joint with John E. Harper), Advances in Mathematics**341**, (2019), 118-187.Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras with the full subcategory of 0-connected O-algebras. This resolves in the affirmative the 0-connected case of a conjecture of Francis-Gaitsgory.

This derived Koszul duality result can be thought of as the spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, and the subsequent p-adic and integral work of Goerss and Mandell on cochains and homotopy type---the following are corollaries of our main result: (i) 0-connected O-algebra spectra are weakly equivalent if and only if their TQ-homology spectra are weakly equivalent as derived K-coalgebras, and (ii) if a K-coalgebra spectrum is 0-connected and cofibrant, then it comes from the TQ-homology spectrum of an O-algebra. We construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups of an O-algebra X, and prove that it converges strongly to the homotopy groups of X when X is 0-connected.

*A nilpotent Whitehead theorem for TQ-homology of structured ring spectra*, (joint with John E. Harper), Tbilisi Mathematical Journal**11(3)**, (2018), 69-79.The aim of this short paper is to prove a TQ-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a TQ-homology analog for structured ring spectra of Dror’s generalized Whitehead theorem for topological spaces; here TQ-homology is short for topological Quillen homology. We also prove retract theorems for the TQ-completion and homotopy completion of nilpotent structured ring spectra.

*Manifolds, K-theory and the calculus of functors*, (joint with Gregory Arone), Contemporary Mathematics,**682**, (2017), 1-38.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 functor is classified by an action of the Koszul dual of the little 3-discs operad.

*Cross-effects and the classification of Taylor towers*, (joint with Gregory Arone), Geometry & Topology,**20-3**, (2016), 1445-1537.We show that the partially-stabilized cross-effects of a functor from either based spaces of spectra, to spectra, admit the structure of a module over a certain operad: the Koszul dual of the little discs operad in the first case, and a desuspension of the commutative operad in the second case. The derivatives inherit a limiting structure as a module over a 'pro-operad'. We show that the Taylor tower of the functor can be recovered from this structure.

*Higher homotopy excision and Blakers-Massey theorems for structured ring spectra*, (joint with John E. Harper), Advances in Mathematics**298**(2016), 654-692.Working in the context of symmetric spectra, we prove higher homotopy excision and higher Blakers-Massey theorems, and their duals, for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We also prove analogous results for algebras and left modules over operads in unbounded chain complexes.

*A classification of Taylor towers for functors of spaces and spectra*, (joint with Gregory Arone), Advances in Mathematics**272**(2015), 471-552.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 construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad. For functors with values in the category of spectra, we give a rather explicit description of the associated comonads and their coalgebras. In particular, for functors from based spaces to spectra we interpret this new structure as that of a divided power right module over the operad formed by the derivatives of the identity on based spaces.

*Coalgebraic models for combinatorial model categories*, (joint with Emily Riehl), Homology, Homotopy and Applications**16**(2014), no. 2, 171-184.We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant.

*Apollonian circle packings of the half-plane*, (joint with John Doyle), Journal of Combinatorics**3**(2012), 1-49.We classify Apollonian circle packings of a half-plane up to Euclidean similarity and describe the relationship between such packings and continued fractions.

*A note on the composition product of symmetric sequences*, Journal of Homotopy and Related Structure**7**(2012), 237-254.We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough structure, namely that of a `normal oplax' monoidal product, to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit of this work is in the dual setting, where it allows us to define a cobar construction for cooperads.

*Bar-cobar duality for operads in stable homotopy theory*, Journal of Topology**5**(2012), 39-80.We describe a bar-cobar duality for operads of spectra. In particular, we construct a Quillen equivalence between the projective model structure on the category of operads, and a new model for the homotopy theory of cooperads of spectra.

*Operads and chain rules for calculus of functors*, (joint with Gregory Arone), Astérisque**338**(2011), 158 pages.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 new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this.

*A chain rule for Goodwillie derivatives of functors from spectra to spectra*, Trans. Amer. Math. Soc.**362**(2010), 399-426.We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor FG at a base object X are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of F at G(X) with the derivatives of G at X. We also consider the question of finding P

_{n}(FG), and give an explicit formula for this when F is homogeneous.*Bar constructions for topological operads and the Goodwillie derivatives of the identity*, Geom. Topol.**9**(2005), 833-933.We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the Lie operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.

### Manuscripts

- Calculus of functors and configuration spaces
This is a summary of a talk given at the Conference on Pure and Applied Topology on the Isle of Skye in June 2005. We describe a relationship between right modules over the operad formed by the Goodwillie derivatives of the identity functor, and configuration spaces on manifolds. The missing step in turning this project into a paper is a proper understanding of the self-Koszul-duality of the Fulton-MacPherson operads of compactified configuration spaces.

- Cross-effects of homotopy functors and spaces of trees
This paper describes models for the cross-effects of homotopy functors given in terms of spaces of trees. These link closely to the methods of the paper "Bar constructions for topological operads and the Goodwillie derivatives of the identity", but I do not have any useful applications for these models.

- Homotopy operations on simplicial algebras over an operad
This was my first attempt as a graduate student to understand what operations exist on the homotopy groups of a simplicial algebra over an operad, in a sense generalizing the Cartan-Bousfield-Dwyer operations on simplicial commutative algebras. It contains some partial results, but not a full description.

## Michael Ching

Department of Mathematics and Statistics

Amherst College

PO Box 5000

Amherst, MA 01002

USA

Telephone: (413) 542-5530

Fax: (413) 542-2550

Email: mching@amherst.edu