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.
We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been implicitly studied. We ask whether the endomorphism operad of the forgetful functor from algebras over an operad to the ground category recovers that operad. The answer is positive for operads in vector spaces over an infinite field, but negative both in vector spaces over finite fields and in sets. Several examples are computed.
Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little $n$-cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of $n$-Poisson algebras in terms of the homology of configuration spaces.
We develop further the theory of operads and analytic functors. In particular, we introduce a bicategory that has operads as 0-cells, operad bimodules as 1-cells and operad bimodule maps as 2-cells, and prove that this bicategory is cartesian closed. In order to obtain this result, we extend the theory of distributors and the formal theory of monads.
We prove that for a topological operad $P$ the operad of oriented cubical chains, $C^{ord}_\ast(P)$, and the operad of singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{ord}_\ast(P;\mathbb{Q})$ is formal if and only if $S_\ast(P;\mathbb{Q})$ is formal, thus linking together some formality results spread in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give different variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets defined by $R$-simplicial differentia…
We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to André Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full.
It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint $Sym_1$ given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author's theory of higher operads, the nonsymmmetric operads are 1-operads and $Sym_1$ is the first term of the infinite series of left adjoint functors $Sym_n,$ called symmetrisation functors, from $n$-operads to symmetric operads with the property that the category of one object, one arrow, . . ., one $…
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…
The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a right adjoint. The condition is a factorization axiom which roughly says that operations in the codomain operad can be written essentially uniquely as operations in arity one followed by operations in the domain operad.
In this article is studied the construction of free operads functor, for the symmetric and non-symmetric case. In order to do this, the operads are seen as monoids on the differential graded modules category. In the last part we show some properties between the associated functors of this construction that may help to better understand the mechanism in this process.
We show that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of infinity-operads to a certain model category of profinite infinity-operads. The construction is based on a notion of lean infinity-operad, and we characterize those infinity-operads weakly equivalent to lean ones in terms of homotopical finiteness properties. Several variants of the construction are also discussed, such as the cases of unital (or closed) infinity-operads and of infinity-categories…
In the theory of operads we consider functors of generalized symmetric powers defined by sums of coinvariant modules under actions of symmetric groups. One observes classically that the construction of symmetric functors provides an isomorphism from the category of symmetric modules to a subcategory of the category of functors on the base category. The purpose of this book is to obtain a similar relationship for functors on a category of algebras over an operad. We observe that right modules over operads, symmetric modules equipped with a right operad action, give rise to functors on categorie…
We study the splitting of the Goodwillie towers of functors in various settings. In particular, we produce splitting criteria for functors $F: \A \to M_A$ from a pointed category with coproducts to $A$-modules in terms of differentials of $F$. Here $A$ is a commutative $S$-algebra. We specialize to the case when $\A$ is the category of $\a$-algebras for an operad $\a$ and $F$ is the forgetful functor, and derive milder splitting conditions in terms of the derivative of $F$. In addition, we describe how triples induce operads, and prove that, roughly speaking, a triple $T$ is naturally equivale…
We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric monoidal $\infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\infty$-operads, as a localization of a presheaf $\infty$-category, and we use this to give a simple proof of the equivalence between Lurie's and Barwick's models for $\infty$-operads.
In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial fun…
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 discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of F are a right module over a certain `pro-operad'. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivia…
We study the subcategory of topological operads $P$ such that $P(0) = *$ (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping sp…
The paper gives an explicit description of the Weiss embedding tower in terms of spaces of maps of truncated modules over the framed Fulton-MacPherson operad.
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.
The goal of this paper is to furnish a literature on Goodwillie calculus for functors defined between categories which derive from chain complexes over a ground field $\Bbbk.$ We characterize homogeneous functors $F: \mathcal{C} \longrightarrow \mathcal{D}$ where $\mathcal{C} ,\mathcal{D}= Ch$ (chain complexes), $Ch_+$(non-negatively graded chain complexes) or $\text{Alg}_\mathcal{O}$ (algebras over a chain complex operad $\mathcal{O}$). In the particular case when $\mathcal{D}= \text{Alg}_\mathcal{O},$ our characterization requires $\Bbbk$ to be of characteristics $0.$ We are then extending t…
The goal of the present paper is to compare, in a precise way, two notions of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk--Weiss in order to generalize the simplicial notion of infinity-category of Joyal--Lurie. This functor extends in two directions the simplicial nerve of Faonte--Lurie for A-infinity-…
Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived Quillen adjunctions of monoidal V-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on M. As an application, we prove that in the category of symmetric spectr…
This paper describes a consequence of the more general results of a previous paper which is of independent interest. We construct a functor from the category of dendroidal sets, which models the theory of infinity-operads, into the category of E-infinity-spaces. Applying May's infinite loop space machine for E-infinity-spaces then gives an infinite loop space machine for infinity-operads. We show that our machine exhibits the homotopy theory of E-infinity-spaces as a localization of the homotopy theory of infinity-operads.
Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural ``fattening'' of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. Co-rings over operads are thus ``relative operads,'' parametrizing morphisms as operads parametrize (co)algebras. Let A denote the associative operad in the category of chain complexes. We de…
For a "genuine" equivariant commutative ring spectrum $R$, $π_0(R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous work we studied equivariant commutative ring structures parametrized by $N_\infty$ operads. In a precise sense, these interpolate between "naive" and "genuine" equivariant ring structures. In this paper, we describe the algebraic analogue of $N_\infty$…
In this paper, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad E_d of little d-dimensional disks, we show that each (d-1)-manifold gives rise to a theory of modules over E_d-algebras and each bordism gives rise to a functor from the category defined by its incoming boundary to the category defined by its outgoing boundary. We describe how to assemble these categories into a map from a certain operad to the operad of (infinity)-categories.
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.
We develop an alternative to the May-Thomason construction used to compare operad based infinite loop machines to that of Segal, which relies on weak products. Our construction has the advantage that it can be carried out in $Cat$, whereas their construction gives rise to simplicial categories. As an application we show that a simplicial algebra over a $Σ$-free $Cat$ operad $\mathcal{O}$ is functorially weakly equivalent to a $Cat$ algebra over $\mathcal{O}$. When combined with the results of a previous paper, this allows us to conclude that up to weak equivalences the category of $\mathcal{O}…
The goal of this memoir is to prove that the bar complex B(A) of an E-infinity algebra A is equipped with the structure of a Hopf E-infinity algebra, functorially in A. We observe in addition that such a structure is homotopically unique provided that we consider unital operads which come equipped with a distinguished 0-ary operation that represents the natural unit of the bar complex. Our constructions rely on a Reedy model category for unital Hopf operads. For our purpose we define a unital Hopf endomorphism operad which operates functorially on the bar complex and which is universal with th…
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…
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…
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…
We generalize Berger and Moerdijk's results on axiomatic homotopy theory for operads to the setting of enriched symmetric monoidal model categories, and show how this theory applies to orthogonal spectra. In particular, we provide a symmetric fibrant replacement functor for the positive stable model structure.
The derivatives of the identity functor on spaces in Goodwillie calculus forms an operad in spectra. Antolin-Camarena computed the mod 2 homology of free algebras over this operad for 1-connected spectra. In this present paper we carry out similar computations for mod p homology for odd primes p.
We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. In the general case of complex schemes with non-pure weight filtration, we relate the singular chains functor…
We propose a $λ$-calculus-style formal language, called the $μ$-syntax, as a lightweight representation of the structure of cyclic operads. We illustrate the rewriting methods behind the formalism by giving a complete step-by-step proof of the equivalence between the unbiased and biased definitions of cyclic operads.
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.
It was shown in a recent paper by Boavida de Brito and Weiss that a well-known construction which to a plain (=monochromatic) topological operad associates a topological category and a functor from it to the category of finite sets is homotopically fully faithful, under mild conditions on the operads. The main result here is a generalization of that statement to k-truncated plain topological operads. A k-truncated operad is a weaker version of operad where all operations have arity at most k.
In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Ho…
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.
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…
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…
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 enrichments. In this paper, we correct this by defining an enriched version of manifold calculus which essentially extends the discrete setting. Along the way, we recast the Taylor tower as a tower of homotopy sheafifications. As a spin-off we obtain a natural connection to operads: the limit of the Taylor tower is a certain (derived) space of right module maps…
