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A filtration of the morphisms of the $k$-linearization $k \mathbf{FS}$ of the category $\mathbf{FS}$ of finite sets and surjections is constructed using a natural $k \mathbf{FI}^{op}$-module structure induced by restriction, where $\mathbf{FI}$ is the category of finite sets and injections. This yields the `primitive' subcategory $ k \mathbf{FS}^0 \subset k \mathbf{FS}$ that is of independent interest. Working over a field of characteristic zero, the subquotients of this filtration are identified as bimodules over $k \mathbf{FB}$, where $\mathbf{FB}$ is the category of finite sets and bijections, also exhibiting and exploiting additional structure. In particular, this describes the underlying $k \mathbf{FB}$-bimodule of $k \mathbf{FS}^0$., Comment: 14 pages. Comments welcome

We study the structure of the category of representations of $\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features. We first construct the simple representations, recovering the classification given by Wiltshire-Gordon. The construction given here also yields explicit descriptions of the indecomposable projectives. These results are used to give a convenient set of projective generators of the category of representations of $\mathbf{FA}$ and hence a Morita equivalence result. This is used to explain how to calculate the multiplicities of the composition factors of an arbitrary object, based only on its underlying $\mathbf{FB}$-representation, where $\mathbf{FB}$ is the category of finite sets and bijections. This is applied to show how to calculate the morphism spaces between projectives in our chosen set of generators, as well as for a closely related family of objects (the significance of which can be shown by relative nonhomogeneous Koszul duality theory)., Comment: v2: Reorganization with some improvements of the exposition. 27 pages. Comments always welcome

The purpose of this paper is to show how Positselski's relative nonhomogeneous Koszul duality theory applies when studying the linear category underlying the PROP associated to a (non-augmented) operad of a certain form, in particular assuming that the reduced part of the operad is binary quadratic. In this case, the linear category has both a left augmentation and a right augmentation (corresponding to different units), using Positselski's terminology. The general theory provides two associated linear differential graded (DG) categories; indeed, in this framework, one can work entirely within the DG realm, as opposed to the curved setting required for Positselski's general theory. Moreover, DG modules over DG categories are related by adjunctions. When the reduced part of the operad is Koszul (working over a field of characteristic zero), the relative Koszul duality theory shows that there is a Koszul-type equivalence between the appropriate homotopy categories of DG modules. This gives a form of Koszul duality relationship between the above DG categories. This is illustrated by the case of the operad encoding unital, commutative associative algebras, extending the classical Koszul duality between commutative associative algebras and Lie algebras. In this case, the associated linear category is the linearization of the category of finite sets and all maps. The relative nonhomogeneous Koszul duality theory relates its derived category to the respective homotopy categories of modules over two explicit linear DG categories., Comment: 52 pages. Comments welcome

This paper exhibits fundamental structure underlying Lie algebra homology with coefficients in tensor products of the adjoint representation, mostly focusing upon the case of free Lie algebras. The main result yields a DG category that is constructed from the PROP associated to the Lie operad. Underlying this is a two-term complex of bimodules over this PROP; it is a quotient of the universal Chevalley-Eilenberg complex. The homology of this DG category is intimately related to outer functors over free groups (introduced in earlier joint work with Vespa). This uses the author's previous results relating functors on free groups to representations of the PROP associated to the Lie operad. This gives a direct algebraic explanation as to why the degree one homology should correspond to an outer functor. Hitherto, the only known argument relied upon the relationship with the higher Hochschild homology functors that arise from the work of Turchin and Willwacher., Comment: 18 pages. Comments welcome

The author has shown that the category of analytic contravariant functors on $\mathbf{gr}$, the category of finitely-generated free groups, is equivalent to the category of left modules over the PROP associated to the Lie operad, working over $\mathbb{Q}$. This exploited properties of the polynomial filtration of the category of contravariant functors on $\mathbf{gr}$. The first purpose of this paper is to strengthen the corresponding result for covariant functors on $\mathbf{gr}$. This involves introducing the appropriate analogue of the category of analytic contravariant functors, namely a certain category of towers of polynomial functors on $\mathbf{gr}$. This category is abelian and has a natural symmetric monoidal structure induced by the usual tensor product of functors. Moreover, the projective generators of this category are described in terms of the Mal'cev functors that are introduced here. It follows that this category is equivalent to the category of right modules over the PROP associated to the Lie operad. As a fundamental example, the Passi functors arising from the group ring functors are described explicitly. The theory is applied to consider bifunctors on $\mathbf{gr}$. This allows the $\mathbb{Q}$-linearization of the category of free groups to be described, up to polynomial filtration. As a stronger application of the theory, this is generalized to the Casimir PROP associated to the Lie operad, as studied by Hinich and Vaintrob. Up to polynomial filtration, this recovers the category $\mathbf{A}$ introduced by Habiro and Massuyeau in their study of bottom tangles in handlebodies., Comment: 43 pages. Comments welcome

This paper is motivated by the study of Turchin and Willwacher's bead representations. The problem is reformulated here in terms of the Lie algebra homology of a free Lie algebra with coefficients in tensor products of the adjoint representation. The main idea is to exploit the truncation of the coefficients given by killing Lie brackets of length greater than two. Although this truncation is brutal, it retains significant and highly non-trivial information, as exhibited by explicit results. A d\'evissage is used that splits the problem into two steps, separating out a `homology' calculation from `antisymmetrization'. This involves some auxiliary categories, including a generalization of the upper walled Brauer category. This approach passes through the `baby bead representations' of the title, for which complete results are obtained. As an application, the composition factors of Turchin and Willwacher's bead representations are calculated for a new infinite family., Comment: 59 pages

The purpose of this paper is to give information on the FI-homology of the standard injective cogenerators of the category of FI-modules, where FI is the category of finite sets and injections. Working over a field k of characteristic zero, a full calculation is given in homological degree zero and a conjectural description in higher homological degree. The proof of the main theorem reduces to a calculation in representation theory of the symmetric groups, exploiting the Young orthonormal basis., Comment: 22 pages

For $\mathcal{O}$ an operad in $k$-vector spaces, the category $\mathcal{F}_\mathcal{O}$ is defined to be the category of $k$-linear functors from the PROP associated to $\mathcal{O}$ to $k$-vector spaces. Given $\mu \in \mathcal{O} (2)$ that satisfies a right Leibniz condition, the full subcategory $\mathcal{F}_\mathcal{O}^\mu \subset \mathcal{F}_\mathcal{O}$ is introduced here and its properties studied. This is motivated by the case of the Lie operad, where $\mu$ is taken to be the generator. By previous results of the author, when $k = \mathbb{Q}$, $\mathcal{F}_{Lie}$ is equivalent to the category of analytic functors on the opposite of the category $\mathbf{gr}$ of finitely-generated free groups. The main result shows that $\mathcal{F}_{Lie}^\mu$ identifies with the category of outer analytic functors, as introduced in earlier work of the author with Vespa. Using this identification, this theory has applications to the study of the higher Hochschild homology functors related to work of Turchin and Willwacher., Comment: v3: minor revision; now 20 pages. v2: updated presentation, with some improvements. Main results unchanged. 18 pages

Working over a field $k$ of characteristic zero, the category of analytic contravariant functors on the category of finitely-generated free groups is shown to be equivalent to the category of representations of the $k$-linear category associated to the Lie operad. Two proofs are given of this result. The first uses the original Ginzburg-Kapranov approach to Koszul duality of binary quadratic operads and the fact that the category of analytic contravariant functors is Koszul. The second proof proceeds by making the equivalence explicit using the $k$-linear category associated to the operad encoding unital associative algebras, which provides the `twisting bimodule'. A key ingredient is the Poincar\'e-Birkhoff-Witt theorem. Using the explicit formulation, it is shown how this equivalence reflects the tensor product on the category of analytic contravariant functors, relating this to the convolution product for representations of the category associated to the Lie operad., Comment: (v2) 34 pages: some improvements to the exposition. (v1) 29 pages

A unified splitting result for Ext calculated in the category of modules over a Leibniz algebra is given for the case where coefficients are either both symmetric modules or both antisymmetric modules. This is a generalization of results of Loday and Pirashvili and others., Comment: 18 pages

Pirashvili exhibited a small subcomplex of the Leibniz complex $(T(s \mathfrak{g}), d_{\mathrm{Leib}})$ of a Leibniz algebra $\mathfrak{g}$. The main result of this paper generalizes this result to show that the primitive filtration of $T(s\mathfrak{g})$ provides an increasing, exhaustive filtration of the Leibniz complex by subcomplexes, thus establishing a conjecture due to Loday. The associated spectral sequence is used to give a new proof of Pirashvili's conjecture that, when $\mathfrak{g}$ is a free Leibniz algebra, the homology of the Pirashvili complex is zero except in degree one. This result is then used to show that the desuspension of the Pirashvili complex carries a natural $L_\infty$-structure that induces the natural Lie algebra structure on the homology of the complex in degree zero., Comment: 21 pages: revised version, following suggestions of a referee (accepted). (v1: 18 pages.)

For $\mathcal{O}$ a reduced operad, a generalized divergence from the derivations of a free $\mathcal{O}$-algebra to a suitable trace space is constructed. In the case of the Lie operad, this corresponds to Satoh's trace map and, for the associative operad, to the double divergence of Alekseev, Kawazumi, Kuno and Naef. The generalized divergence is shown to be a $1$-cocycle for the usual Lie algebra structure on derivations. These results place the previous constructions into a unified framework; moreover, they are natural with respect to the operad. An important new ingredient is the use of naturality with respect to the category of finite-rank free modules and split monomorphisms over a commutative ring $R$. This allows the notion of torsion for such functors to be exploited. Supposing that the ring $R$ is a PID and that the operad $\mathcal{O}$ is binary, the main result relates the kernel of the generalized divergence to the sub Lie algebra of the Lie algebra of derivations that is generated by the elements of degree one with respect to the grading induced by arity., Comment: 41 pages

Pirashvili's Dold-Kan type theorem for finite pointed sets follows from the identification in terms of surjections of the morphisms between the tensor powers of a functor playing the role of the augmentation ideal; these functors are projective. We give an unpointed analogue of this result: namely, we compute the morphisms between the tensor powers of the corresponding functor in the unpointed context. We also calculate the Ext groups between such objects, in particular showing that these functors are not projective; this is an important difference between the pointed and unpointed contexts. This work is motivated by our functorial analysis of the higher Hochschild homology of a wedge of circles.

Working over the prime field F_p, the structure of the indecomposables Q^* for the action of the algebra of Steenrod reduced powers A(p) on the symmetric power functors S^* is studied by exploiting the theory of strict polynomial functors. In particular, working at the prime 2, representation stability is exhibited for certain related functors, leading to a conjectural representation stability description of quotients of Q^* arising from the polynomial filtration of symmetric powers., Comment: Revised version (with modified title), focusing upon the core material; now 20 pages

We study higher Hochschild homology evaluated on wedges of circles, viewed as a functor on the category of free groups. The main results use coefficients arising from square-zero extensions; this is motivated by work of Turchin and Willwacher in relation to hairy graph cohomology. The functorial point of view allows us to exploit tools such as the theory of polynomial functors and exponential functors. We also introduce and make essential use of the category of outer functors, the full subcategory of functors on free groups on which inner automorphisms act trivially. We give a description of higher Hochschild homology in terms of intrinsically defined polynomial outer functors; we also obtain several explicit computations of these outer functors, working over a field of characteristic zero. In particular, higher Hochschild homology gives a natural source of non-trivial polynomial outer functors., Comment: Major revision; To appear in Bulletin de la SMF

Inspired by the work of Henn, Lannes and Schwartz on unstable algebras over the Steenrod algebra modulo nilpotents, a characterization of unstable algebras that are $A$-finitely generated up to nilpotents is given in terms of the associated presheaf, by introducing the notion of a finite presheaf. In particular, this gives the natural characterization of the (co)analytic presheaves that are important in the theory of Henn, Lannes and Schwartz. However, finite presheaves remain imperfectly understood, as illustrated by examples. One important class of examples is shown to be provided by unstable algebras of finite transcendence degree (under a necessary weak finiteness condition). For unstable Hopf algebras, it is shown that the situation is much better: the associated presheaf is finite if and only if its growth function is polynomial. This leads to a description of unstable Hopf algebras modulo nilpotents in the spirit of Henn, Lannes and Schwartz., Comment: Slightly streamlined version, now 24 pages

Working over the prime field of characteristic two, consequences of the Koszul duality between the Steenrod algebra and the big Dyer-Lashof algebra are studied, with an emphasis on the interplay between instability for the Steenrod algebra action and that for the Dyer-Lashof operations. The central algebraic framework is the category of length-graded modules over the Steenrod algebra equipped with an unstable action of the Dyer-Lashof algebra, with compatibility via the Nishida relations. A first ingredient is a functor defined on modules over the Steenrod algebra that arose in the work of Kuhn and McCarty on the homology of infinite loop spaces. This functor is given in terms of derived functors of destabilization from the category of modules over the Steenrod algebra to unstable modules, enriched by taking into account the action of Dyer-Lashof operations. A second ingredient is the derived functors of the Dyer-Lashof indecomposables functor to length-graded modules over the Steenrod algebra. These are related to functors used by Miller in his study of a spectral sequence to calculate the homology of an infinite delooping. An important fact is that these functors can be calculated as the homology of an explicit Koszul complex with terms expressed as certain Steinberg functors. The latter are quadratic dual to the more familiar Singer functors. By exploiting the explicit complex built from the Singer functors which calculates the derived functors of destabilization, Koszul duality leads to an algebraic infinite delooping spectral sequence. This is conceptually similar to Miller's spectral sequence, but there seems to be no direct relationship. The spectral sequence sheds light on the relationship between unstable modules over the Steenrod algebra and all modules., Comment: Minor revision with no changes to mathematical content; 36 pages (v1 38 pages)

Let $R_s M$ denote the Singer construction on an unstable module $M$ over the Steenrod algebra $A$ at the prime two; $R_s M$ is canonically a subobject of $P_s\otimes M$, where $P_s$ is the polynomial algebra on s generators of degree one. Passage to $A$-indecomposables gives the natural transformation $R_s M \rightarrow F \otimes_A (P_s \otimes M)$, which identifies with the dual of the composition of the Singer transfer and the Lannes-Zarati homomorphism. The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first named author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one., Comment: v2 15 pages. Minor revision. v3 17 pages, revision following referee's recommendations. Accepted for publication J. Alg

Double Poisson structures (a la Van den Bergh) on commutative algebras are studied; the main result shows that there are no non-trivial such structures on polynomial algebras of Krull dimension greater than one. For a general commutative algebra A, this places significant restrictions on possible double Poisson structures. Exotic double Poisson structures are exhibited by the case of the polynomial algebra on a single generator, previously considered by Van den Bergh., Comment: 12 pages; very minor revision

The spectral sequence associated to the Arone-Goodwillie tower for the n-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod 2 Steenrod algebra, together with properties of a generalized class of almost unstable modules which is introduced here. An essential ingredient of the proof is a non-vanishing result for certain extension groups in the category of unstable modules localized away from nilpotents., Comment: Minor revision. Sketch of case p odd added. This version 27 pages

These notes explain how to construct small functorial chain complexes which calculate the derived functors of destabilization (respectively iterated loop functors) in the theory of modules over the mod 2 Steenrod algebra; this shows how to unify results of Singer and of Lannes and Zarati., Comment: Minor revision; more detail provided on the Lannes-Zarati morphism and a new section giving perspectives. 32 pages. (v2 Minor revision. 25 pages.)

A structure theorem for bounded-below modules over the subalgebra A(1) of the mod 2 Steenrod algebra generated by Sq^1, Sq^2 is proved; this is applied to prove a classification theorem for a family of indecomposable A(1)-modules. The action of the A(1)-Picard group on this family is described, as is the behaviour of duality. The cohomology of dual Brown-Gitler spectra is identified within this family and the relation with members of the A(1)-Picard group is made explicit. Similarly, the cohomology of truncated projective spaces is considered within this classification; this leads to a conceptual understanding of various results within the literature. In particular, a unified approach to Ext-groups relevant to Adams spectral sequence calculations is obtained, englobing earlier results of Davis (for truncated projective spaces) and recent work of Pearson (for Brown-Gitler spectra)., Comment: Revision - mathematical content essentially unchanged (39 pages)

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod $p$ singular cohomology of a space, for $p$ a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes' mapping space theorem. The second (for the prime $2$) is based on an analysis of the $-1$ and $-2$ columns of the Eilenberg-Moore spectral sequence, and of the associated extension. In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between $\Fp$-vector spaces. The second result in particular exhibits the power of the functorial approach.

The structure of the BP-cohomology of elementary abelian p-groups is studied, obtaining a presentation expressed in terms of BP-cohomology and mod-p singular cohomology, using the Milnor derivations. The arguments are based on a result on multi-Koszul complexes which is related to Margolis's criterion for freeness of a graded module over an exterior algebra., Comment: Minor revision (18 pages)

A general notion of detection is introduced and used in the study of the cohomology of elementary abelian 2-groups with respect to the spectra in the Postnikov tower of orthogonal K-theory. This recovers and extends results of Bruner and Greenlees and is related to calculations of the (co)homology of the spaces of the associated Omega-spectra by Stong and by Cowen Morton.

The connective ku-(co)homology of elementary abelian 2-groups is determined as a functor of the elementary abelian 2-group. The argument requires only the calculation of the rank one case and the Atiyah-Segal theorem for KU-cohomology together with an analysis of the functorial structure of the integral group ring. The methods can also be applied to the odd primary case. These results are used to analyse the local cohomology spectral sequence calculating ku-homology, via a functorial version of local duality for Koszul complexes. This gives a conceptual explanation of results of Bruner and Greenlees.

The notion of quadratic self-duality for coalgebras is developed with applications to algebraic structures which arise naturally in algebraic topology, related to the universal Steenrod algebra via an appropriate form of duality. This explains and unifies results of Lomonaco and Singer., Comment: 18 pages

An explicit chain complex is constructed to calculate the derived functors of destabilization at an odd prime, generalizing constructions of Zarati and of Hung and Sum. The methods are based on the ideas of Singer and Miller and also apply at the prime two. A structural result on the derived functors of destabilization is deduced., Comment: Significant revision (27 pages); accepted for publication in Acta Mathematica Vietnamica (AMV)

The purpose of this paper is to investigate an algebraic version of the double complex transfer, in particular the classes in the two-line of the Adams-Novikov spectral sequence which are the image of comodule primitives of the MU-homology of the product of two copies of infinite complex projective space via the algebraic double transfer. These classes are analysed by two related approaches; the first, p-locally for an odd prime, by using the morphism induced in MU-homology by the chromatic factorization of the double transfer map together with the f'-invariant of Behrens (for p>=5). The second approach uses the algebraic double transfer and the f-invariant of Laures., Comment: 25 pages (revised)

Lannes' T-functor is used to give a construction of the Singer functor R_1 on the category U of unstable modules over the Steenrod algebra A. This leads to a direct proof that the composite functor Fix R_1 is naturally equivalent to the identity. Further properties of the functors R_1 are deduced, especially when applied to reduced and nilclosed unstable modules.

A generalization of a theorem of Crabb and Hubbuck concerning the embedding of flag representations in divided powers is given, working over an arbitrary finite field F, using the category of functors from finite-dimensional F-vector spaces to F-vector spaces., Comment: Accepted for publication in JHRS

This paper extends the relation established for group cohomology by Green, Hunton and Schuster between chromatic phenomena in stable homotopy theory and certain natural subrings of singular cohomology. This exploits the theory due to Henn, Lannes and Schwartz of unstable algebras over the Steenrod algebra localized away from nilpotents., Comment: This is the version published by Algebraic & Geometric Topology on 24 August 2006

A new and natural description of the category of unstable modules over the Steenrod algebra as a category of comodules over a bialgebra is given; the theory extends and unifies the work of Carlsson, Kuhn, Lannes, Miller, Schwartz, Zarati and others. Related categories of comodules are studied, which shed light upon the structure of the category of unstable modules at odd primes. In particular, a category of bigraded unstable modules is introduced; this is related to the study of modules over the motivic Steenrod algebra., Comment: This is the version published by Geometry & Topology Monographs on 14 November 2007