Duncan A. Clark September 30, 2020 Department of Mathematics, Ohio State University 231 W 18th Ave, Columbus, OH, USA, 43210 email: [email protected] website: https://www.asc.ohio-state.edu/clark.1843/home

Research Statement My research concerns describing the structure inherent to the Goodwillie derivatives of homotopy preserving functors between suitable categories (for instance, based topological spaces Top∗, spectra Spt, or more a general model category C). Introduced by Goodwillie in a landmark series of papers [34], [35], [36], homotopy functor calculus is one of several flavors of “calculus” [59], [44], [49, §6] which have spurred a wide range of work within homotopy theory and related fields in the last few decades (see [4] for a recent survey). The main idea is to approximate a functor F between such suitable categories by a tower of n-excisive or “polynomial of degree at most n” functors, PnF , of the form

F → · · · → PnF → Pn−1F → · · · → P1F → P0F.

These functors PnF play a role analogous to the Taylor polynomials pnf associated to an infinitely differentiable function f : R → R in ordinary calculus and analysis. Suppose that F is a functor Top∗ → Top∗. Goodwillie shows in [36] that the formula (n) n pnf(x) − pn−1f(x) = f (0)x /n! has a homotopical analog in that the “difference” (in this case, homotopy fiber) between PnF and Pn−1F is measured by a certain spectrum ∂nF (the n-th derivative of F ) as an equivalence of the form ∞ ∞ ∧n DnF (X) = hofib(PnF (X) → Pn−1F (X)) ' Ω (∂nF ∧ Σ X )hΣn . Moreover, under suitable niceness conditions on F , this Taylor tower will converge when evaluated on spaces X whose homotopy groups are also sufficiently nice; i.e., F (X) ' holimn→∞ PnF (X). The n-th derivative ∂nF comes equipped with an action by the n-th symmetric group Σn, and the subscript hΣn in the above formula for DnF (X) indicates homotopy orbits are taken with respect to this action along with the permutation action on Σ∞X∧n. The notation Σ∞X denotes the suspension spectrum of a space X, and ∧ denotes the smash product of spectra (we will define these more rigorously in the following section). Recent work. My recent work has dealt with describing structure inherent to the derivatives of the identity functor in various model categories C. Specifically, when C = AlgO is a category of structured ring spectra described as an algebra over an operad O in spectra, I have shown [22] that the derivatives of the identity can be given a “highly homotopy coherent operad” which is equivalent to O, thus answering a long-standing conjecture in the field1. I have designed my techniques to be implemented quite generally, and in [23] have further provided a new description of an operad structure for the derivatives of the identity functor in based topological spaces, as first described by Ching [17]. The key component in these results is to utilize the box product  of [8] to induce the desired operad structure by working on the level of cosimplicial resolutions. I have used these box product pairings to further show [23] that the “derived primitives” of certain coalgebras in spectra inherit a natural action by the derivatives of the identity in spaces. One aim of my current project is to use similar techniques to provide a comparison between a suitable model category C and algebras over ∂∗IC—a main part of which is showing the latter admits a natural operad structure [19].

1Though attributed to Haynes Miller, this conjecture can be found written in [1] and [54]. 1 2

Spectra and the Taylor tower of the identity in spaces

A spectrum is a sequence of based spaces Y = {Yn}n≥0 connected by continuous maps ΣYn → 1 2 3 Yn+1 (here ΣYn is the suspension S ∧Yn, e.g. ΣS = S ) which may be thought of as a topological analog to a chain complex of abelian groups. As with chain complexes, spectra come equipped with their own invariants, stable homotopy groups

πi(Y ) = colimn→∞ πi+n(Yn), (−∞ ≤ i ≤ ∞) built from the homotopy groups of the spaces Yn. In particular, any pointed space X gives rise to ∞ ∞ n a suspension spectrum Σ X defined by (Σ X)n = S ∧ X. s The stable homotopy groups of a space X—denoted π∗(X)—are simply the stable homotopy ∞ groups π∗(Σ X) and may be thought of as a “first order approximation” to the ordinary homotopy groups of X as made precise Goodwillie’s homotopy functor calculus. The assignment X 7→ Σ∞X ∞ ∞ admits a right adjoint Ω defined on certain “nice” spectra (i.e., Ω-spectra) as Ω Y = Y0. As ∞ ∞ with any adjunction, there is a unit map ITop∗ → Ω Σ which in this case induces a space-level s comparison map between ordinary homotopy groups π∗ and stable homotopy groups π∗. Linear approximations to functors of spaces. It is precisely this stable homotopy functor Ω∞Σ∞ which appears as the first term in the Taylor tower of the identity on based spaces. Notably this implies that ITop∗ is not linear. In fact, linearity of the identity is a reflection of the model category in question being stable; i.e., that suspension X 7→ ΣX admit a “homotopy inverse” 1 X 7→ ΩX = Map∗(S ,X)

(in Top∗ this is the based loop space on X). The Freudenthal suspension theorem [30] provides isomorphisms πi(X) → πi(ΩΣX) = πi+1(ΣX) for a “stable range” of i, though notably not for all i (for instance, take X = S2). Comparatively, there is always an isomorphism in homology ∼ Hi(X) = Hi+1(ΣX) obtained via the Mayer-Vietoris sequence. The category Spt of spectra may then be thought of as the “best” approximation to Top∗ by a stable category. Moreover, the linearization P1ITop∗ is the best approximation to ITop∗ by a linear functor on Spt (i.e., one of the form X 7→ E ∧ X for some fixed spectrum E) which in this case is just the identity ISpt. More generally, for any homotopy functor F : Top∗ → Top∗ such that F (∗) ' ∗, the linear approximation P1F is equivalent to the stabilization of F ; i.e., ∞ ∞ n n P1F ' Ω F Σ = colimn Ω F Σ . Furthermore, any such linear functor represents a homology theory (i.e., satisfies the Eilenberg- Steenrod axioms [25]).

Example: The Taylor tower of the identity in spaces. The Taylor tower of ITop∗ is well understood and plays several key roles in better understanding the homotopy theory of based topological spaces. The higher Blakers-Massey theorems [26], [35] show that ITop∗ will converge on simply- connected spaces. Work of [45], [6], [5] describes the layers DnITop∗ and derivatives ∂nITop∗ as the Spanier-Whitehead duals of certain partition poset complexes Par(n); i.e., ∞ ∧n DnITop∗ (X) ' Map(Par(n), Σ X )hΣn , ∂nITop∗ ' Map(Par(n),S), (n ≥ 0).

Ching uses these models to construct an operad structure on derivatives of the ITop∗ in [17] as a generalization of the bar-cobar duality of [32] to topological operads. This operad plays a central role in understanding the structure implicit to the derivatives of a general homotopy functor between Top∗ or Spt [1], [2], and in describing homotopy descent data for certain “chromatic localizations” of spaces [42], [10] (we will return to these points later). It is expected that an operad structure on ∂∗IC should exist for a wide class of suitable model categories C. For instance, I have shown this is indeed so when C is a category of algebras over an operad in spectra, the necessary background of which is summarized in the following sections. 3

Operadic algebras in spectra A major result of modern homotopy theory is the construction of a point-set model for a sym- metric monoidal smash product ∧ = ⊗S of spectra [27], [43], [50] (whose unit is the sphere spectrum S = Σ∞S0) which acts as a homotopy-theoretic analog to the tensor product of abelian groups. In analogy with a ring R being an abelian group with a suitable multiplication R ⊗ R → R and unit, a ring spectrum is a spectrum X which admits suitable maps of the form X ∧ X → X (multiplication) S → X (unit) It is often the case that such maps can only be made associative or commutative up to coherent homotopies which can naturally be encoded an algebra over an operad [12], [13], [52]. For instance, the Eilenberg-MacLane spectrum HR on a commutative ring R is a commonly employed spectrum n as it represents R-cohomology via He (X; R) = [X,HRn] (here, [−, −] denotes homotopy classes of maps). Here, HRn is an Eilenberg-MacLane space K(R, n) (i.e., πkK(R, n) = ∗ for k 6= n, and πnK(R, n) = R) for all n. Notably HR admits a multiplication HR ∧ HR → HR, induced by that on R, which can at best be made “highly homotopy commutative”—that is to say, HR is an E∞ ring spectrum [51]. E∞ ring spectra are of great interest to homotopy theorists; for instance, there are certain “power operations” induced on the cohomology theories which they represent [56]. Operadic algebras. Operads are useful tools for describing operations which may be coherent only up to higher homotopies, such as the structure found on an n-fold loop space [52]. Let us write O for an operad in spectra. An O-algebra consists of a spectrum X together with structure maps of the form ∧n O[n] ∧Σn X → X, (n ≥ 0) The term O[n] is a spectrum with an action by Σn which parametrizes n-ary operations on X. For instance, the commutative operad is given by Com[n] = S for all n ≥ 0 (with trivial Σn action), and Com-algebras are those spectra X which admit associative multiplication maps X∧n → X ∧n equivariant with respect to the permutation action on X . We write AlgO for the category of such algebras over a fixed operad O in spectra such that O[0] = ∗ (which guarantees that AlgO is pointed). In particular, our O-algebras will be non-unital, a condition which naturally arises when studying augmented commutative structures. Stabilization and the Taylor tower of the identity for operadic algebras. The category AlgO inherits a model structure from that on Spt [55], [37], and further admits a simplicial ten- soring which allows one to build its stabilization as the category of Bousfield-Friendlander spectra Spt(Alg ) [16], [57]. It is known in this context [7], [54] that the “suspension spectrum” Σ∞ X is O AlgO equivalent to the topological Quillen homology spectrum TQ(X). TQ is the closest homotopy-meaningful replacement (i.e., derived functor) of Q, where Q(X) is obtained from X ∈ AlgO by destroying all 2-ary and higher operations prescribed by O, and may be thought of as a homotopical analog to the usual indecomposables quotient from commutative algebra. Furthermore, Q admits a right adjoint U which prescribes a spectrum Y with trivial n-ary operations from O, and the composite UQ (suitably derived) naturally arises as the linear approximation P1IAlgO . More generally, Harper-Hess provide a description of the Taylor tower of

IAlgO as the homotopy completion tower associated to an O-algebra X [39], [54]. For n ≥ 2, the n-th polynomial approximation PnIAlgO (X) is obtained from X by destroying only those operations prescribed by O for levels n + 1 and higher, and the layer DnIAlgO is further determined by O[n] in that ∧n DnIAlgO (X) ' U(O[n] ∧ TQ(X) )hΣn .

The Taylor tower for IAlgO may then be thought of as a homotopical analog to the completion tower associated to a non-unital commutative ring R · · · → R/Rn+1 → R/Rn → · · · → R/R3 → R/R2 → ∗. 4

Here, R/Rn denotes the quotient of R by the image of the multiplication map R⊗n → R. In particular, when evaluated on a connective O-algebra X, the Taylor tower will converge, i.e., such X is equivalent to its homotopy completion Xh∧ [39], [20], [11].

A new approach to an operad structure for the derivatives of the identity My technique is to resolve the identity functor via the standard cosimplicial resolution arising from the stabilization adjunction (Q, U) for connective operadic algebras in spectra. Specifically, I obtain a sequence of cosimplicial spectra C(O)[n] along with equivalences

∂nIAlgO ' holim∆ C(O)[n], (n ≥ 0) which provide a useful point-set model for the derivatives of the identity in terms of totalizations. This resolution C(O) is analogous to the diagram which provides the Bousfield-Kan R-completion of a based space [14], [33, VII.3], and arises similarly to the model for ∂nITop∗ described in [5]. Working on the level of resolutions, I construct a monoidal pairing C(O)C(O) → C(O) with respect to the box product  of cosimplicial objects [8], which induces a “highly homotopy- coherent” operad structure on the derivatives of the identity after passing to totalization. This adds to a growing list of recent applications of the box product to induce homotopy coherent compositions [53], [2], [21], and provides an easy to use, algebraic alternative to the operad structure on the derivatives of the identity pioneered in [17]. As such, one technically challenging aspect to my work lies in precisely describing the homotopy coherence of the evident operad structure on the derivatives of the identity. I have developed a specialized category of N-colored operads with levels [22, §6] which is designed to encode and allow for comparisons between operads and “highly homotopy-coherent” operads. The upshot is that within this framework of operads I can demonstrate an equivalence as operads between O and ∂∗IAlgO . Conjecture. We conjecture that similar techniques may to describe an operad structure on the derivatives of the identity in a suitable model category C. Furthermore, we expect algebras over this operad structure to recover the homotopy theory of the subcategory of C of objects such that the Taylor tower of IC converges.

Example: Spectral Lie algebras. Arone-Kankaanrinta observe in [5] that the models for DnITop∗ and ∂nITop∗ from [45], [6] similarly arise from analyzing the Bousfield-Kan cosimplicial resolution ∞ ∞ of ITop∗ with respect to the stabilization adjunction (Σ , Ω ) between Top∗ and Spt via the Snaith splitting. For X ∈ Top∗, the Snaith splitting [58], [24] provides an equivalence a Σ∞Ω∞Σ∞(X) ' Σ∞X∧k k≥1 Σk and may be interpreted to say the Taylor tower for the counit Σ∞Ω∞ splits as a coproduct of its ∞ ∞ homogeneous layers when evaluated on a suspension spectrum. In particular, ∂k(Σ Ω ) ' S with ∞ ∞ trivial Σk action, for all k ≥ 1. As shown in [1], the collection of derivatives ∂∗(Σ Ω ) inherits a ∞ ∞ cooperad structure (essentially the dual of an operad), and moreover that ∂∗(Σ Ω ) is equivalent to the commutative cooperad. Furthermore, ∂∗ITop∗ is equivalent to the cobar construction [17] on ∞ ∞ ∂∗(Σ Ω ) with respect this cooperad structure. This equivalence is an instance of Koszul duality

[32], [31], [29] for operads of spectra. As a consequence, the operad ∂∗ITop∗ is often referred to as the spectral Lie operad, as the Lie operad and commutative cooperad are Koszul dual in the classic setting of rational chain complexes [32]. ∞ A comparison Top∗ → ∂∗ITop∗ -algebras is obtained as follows. Any suspension spectrum Σ X admits a commutative “Tate”-coalgebra structure [41], and such coalgebras admit “derived primi- tives” functor (denoted Prim and dual in construction to TQ), which lands in ∂∗ITop∗ -algebras. This may be thought of as an homotopy theoretic analog to the statement from commutative algebra 5 that the primitives of a Hopf algebra naturally form a Lie algebra. Heuts [41], [42] has further shown that certain chromatic localizations of the ∞-category of simply-connected spaces (i.e., lo- calizations with respect to equivalences on vn-periodic homotopy groups [9]) are equivalent to the category of spectral Lie algebras in T (n)-local spectra via the assignment X 7→ Prim(Σ∞X).

Outline of approach Within my framework, I am able to describe this duality via box product pairings of underlying cosimplicial objects [23]. The hope is that the above strategy can be implemented in general to produce an operad structure on the derivatives of the identity in a suitable model category C.

An operad structure for the derivatives of the identity. In general, it is expected that ∞ ∞ ∂∗(ΩC ΣC ) be a cooperad (at least in the homotopy category) and ∂∗IC be given by the cobar ∞ ∞ complex on this cooperad. The trouble is that the evident cooperadic structure on ∂∗(ΩC ΣC ) is often not rigid enough to precisely construct a cobar resolution. The benefit to my approach we eschew the need for a strict cooperad structure and only require that ∂∗IC be built as the ˚ totalization of a cosimplicial diagram—call it C(IC)—which admits a -monoid structure. This would have to be built by hand, but should be possible given a suitable “Snaith splitting” ∞ ∞ of the counit ΣC ΩC . This is essentially our approach from [22] outlined above for the case of C = Alg , wherein we use a specific model for iterates of stabilization UQ = Ω∞ Σ∞ provided O AlgO AlgO by [48] to obtain the diagram C(O).

The comparison map. A theoretical comparison Ξ from C to ∂∗IC-algebras in Spt(C) is then ∞ ∞ ∞ obtained as follows. The suspension spectrum ΣC X for X ∈ C is naturally a ΣC ΩC -coalgebra. ∞ ∞ Such coalgebras should then be equivalent to Tate-coalgebras over ∂∗(ΣC ΩC ) by a suitable Snaith splitting in C. Again, this cooperad structure may be only defined up to homotopy, but we only ∞ need that the resulting cosimplicial resolution C(ΣC X) (which computes the derived primitives) ˚ admits a -module pairing over C(IC). The map Ξ is then given by ∞ X 7→ Tot C(ΣC X). It is likely that for Ξ to be an equivalence we need to restrict to the subcategory of C such that the stabilization adjunction is comonadic [40]; we conjecture this is so precisely when the Taylor tower ∞ of IC(X) converges. One possible approach is to show that C(ΣC X) admits a coaugmentation (perhaps after localization) which we could show is an equivalence. In AlgO this coaugmentation is the unit X → UQ(X); in Top∗ it is provided by the Bousfield-Kuhn functor [47], [15], [42].

Application: A “highly homotopy coherent” chain rule. An additional application which should be fairly immediate is a generalization of the chain rule [18], [1], [46], [60], [49, §6.3], to categories of operadic algebras, or more generally to functors of suitable model categories. The chain rule map above may be further thought of as an analog of the Fa`adi Bruno formula [28] of ordinary calculus (which provides a formula for the n-th derivative of a composite f◦g in terms of the derivatives of f and g) expressed as a map

∂∗F ◦ ∂∗G → ∂∗(FG) Here, F,G are composable functors and ◦ denotes the “composition” of such sequences [55], [37]. By analogous techniques as before, this map should be induced by a box product pairing of suitable underlying cosimplicial resolutions of ∂∗F and ∂∗G (for instance, as appropriately-sided cobar resolutions [38], [1]). Setting F or G to be the identity would then prescribe additional structure intrinsic to the derivatives of an arbitrary functor; for instance, it is shown in [2], [3] that the Taylor tower of functors between Top∗ or Spt can be rebuilt from this bimodule structure on their derivatives. 6

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