Cacti and filtered distributive laws

A tree is an acyclic connected graph and a rooted tree is a tree with a chosen vertex, the root. A rooted tree may be directed: every edge ${\left\{v,w\right\}}$ may be oriented to $\overrightarrow{vw}$ in such a way that the minimal path from $w$ to the chosen vertex contains ${\left\{v,w\right\}}$. By the acyclicity of the tree this must hold for exactly one of the choices $\overrightarrow{vw}$ and $\overrightarrow{wv}$. The edges may be seen to be directed ‘away from the roots’. We denote by $E(T)$ the set of edges of a tree $T$.

Suppose that $T$ is a tree with vertex set $V$. Let $\mathbf{Y}=\left(Y_{i}\right)_{i\in V}$ be a $V$-tuple of topological spaces. Then a $\mathbf{Y}$-tree is a rooted tree with an edge labelling where the edge $\overrightarrow{ij}$ is labelled by an element of $Y_{i}$. For a space $Y$ as shorthand we define a $Y$-tree to be a $\mathbf{Y}$-tree where the $V$-tuple $\mathbf{Y}$ is constantly $Y$. Then the edge labelling is a map from the edge set $E$ to the space $Y$. Meanwhile a $Y$-forest is a $\mathbf{Y}$-tree where $\mathbf{Y}$ is the $V$-tuple with $Y_{0}\cong{\left\{\bullet\right\}}$, where $0$ is the root vertex and $Y_{v}\cong Y$ for any other vertex. The naming makes sense because by removing the root $0$ and all adjacent vertices we are left with a disjoint union of $Y$-trees; the root of each tree is the unique vertex adjacent to $0$ and the edge labelling is inherited.

To a rooted tree $T$ we define the level $l(T)$ to be the number of non-trivial directed paths in $T$. So for a corolla with root $1$ and $k-1$ other vertices the level is $k-1$, for a tree with root $1$ and edges $\overrightarrow{i(i+1)}$ for $i=1,\ldots,k-1$ the level is $k(k-1)/2$. The level allows one to filter the set of $\mathbf{Y}$-trees.

A coalgebra is an object $C$ of $\mathcal{C}$ equipped with a comultiplication $\Delta\colon C\to C\otimes C$ and a counit $\epsilon\colon C\to\mathbb{I}$ satisfying the conventional coassociativity and counit axioms. For the comultiplication, we often use Sweedler’s notation $\Delta(c)=\sum c_{(1)}\otimes c_{(2)}$. An augmented coalgebra is a coalgebra $C$ equipped with a coalgebra homomorphism $\gamma\colon\mathbb{I}\to C$ such that $\epsilon\gamma=1$. A cocommutative coalgebra is a coalgebra satisfying $\sigma\Delta=\Delta$. Our main focus will be on graded augmented cocommutative coalgebras, that is augmented cocommutative coalgebras in $\operatorname{gVect}$. The main source of such coalgebras relevant for our purposes is topology: the homology coalgebra of a pointed topological space $(Y,\bullet)$ is a graded augmented cocommutative coalgebra. An augmented coalgebra in $\operatorname{Vect}$ or $\operatorname{gVect}$ naturally splits into a direct sum of vector spaces $C=\mathbbold{k}\operatorname{\mathbbold{1}}\oplus\overline{C}$, where $\operatorname{\mathbbold{1}}=\gamma(1)$, $\overline{C}=\ker(\epsilon)$.

For details on operads we refer the reader to the book [25], for details on Gröbner bases for operads — to the paper [13]. In this section we only recall the key notions used throughout the paper. By an operad (enriched in a symmetric monoidal category $(\mathcal{C},\otimes,\sigma,\mathbb{I})$) we mean a monoid in one of the two monoidal categories: the category of symmetric $\mathcal{C}$-collections equipped with the composition product or the category of nonsymmetric $\mathcal{C}$-collections equipped with the shuffle composition product. The former kind of monoids is referred to as symmetric operads, the latter — as shuffle operads. We always assume that our collections are reduced, that is, have no elements of arity $0$. A good rule of thumb is that all operads defined in this paper are symmetric operads, but for computational purposes it is useful to treat them as shuffle operads. This does not lose any information except for the symmetric group actions, since the forgetful functor $\mathrsfs{O}\mapsto\mathrsfs{O}^{f}$ is monoidal and one-to-one on objects (and therefore for tasks that can be formulated without the symmetric group actions, e.g. computing bases and dimensions of components, proving the Koszul property etc., we can choose arbitrarily whether to work with a symmetric operad or with its shuffle version). In the “geometric” setting, $\mathcal{C}$ will usually be the category of sets, or topological spaces, or pointed topological spaces, in the “linear” setting — the category of vector spaces (in which case symmetric collections are usually called $\mathbb{S}$-modules), or the category of graded vector spaces or chain complexes (in which case symmetric collections are called differential graded $\mathbb{S}$-modules). A linear symmetric operad can also be thought as of collection of spaces of operations of some type, and therefore can be defined via its category of algebras, i.e. vector spaces where these operations act, via identities between operations acting on a vector space.

In the linear setting, a very useful technical tool for dealing with (shuffle) operads is given by Gröbner bases. More precisely, similarly to associative algebras, operads can be presented via generators and relations, that is as quotients of free operads $\mathrsfs{F}(\mathrsfs{V})$, where $\mathrsfs{V}$ is the space of generators. The free shuffle operad generated by a given nonsymmetric collection admits a basis of “tree monomials” which can be defined combinatorially; a shuffle composition of tree monomials is again a tree monomial. In addition to the “arity” of elements of a free operad, there is the notion of weight, similar to grading for associative algebras: we define the weight of a tree monomial as the number of generators used in this tree monomial. Weight is well behaved under composition: when composing several tree monomials, the weight of the result is equal to the sum of their weights. For an arbitrary operad $\mathrsfs{O}=\mathrsfs{F}(\mathrsfs{V})/(\mathrsfs{R})$ whose relations $\mathrsfs{R}$ are weight-homogeneous, the weight descends from the free operad $\mathrsfs{F}(\mathrsfs{V})$ on $\mathrsfs{O}$; the subcollection of $\mathrsfs{O}$ consisting of all elements of weight $k$ is denoted by $\mathrsfs{O}_{(k)}$.

There exist several ways to introduce a total ordering of tree monomials in such a way that the operadic compositions are compatible with that total ordering. There is also a combinatorial definition of divisibility of tree monomials that agrees with the naive operadic definition: one tree monomial occurs as a subtree in another one if and only if the latter can be obtained from the former by operadic compositions. A Gröbner basis of an ideal $I$ of the free operad is a system $S$ of generators of $I$ for which the leading monomial of every element of the ideal is divisible by one of the leading terms of elements of $S$. Such a system of generators allows to perform “long division” modulo $I$, computing for every element its canonical representative. There exists an algorithmic way to compute a Gröbner basis starting from any given system of generators (“Buchberger’s algorithm for shuffle operads”).

A part of the operad theory which provides one of the most useful known tools to study homological and homotopical algebra for algebras over the given operad is the Koszul duality for operads [17]. Proving that a given operad is Koszul instantly provides a minimal resolution for this operad, gives a description of the homology theory and, in particular, the deformation theory for algebras over that operad etc. There are a few general methods to prove that an operad is Koszul; one of the simplest and widely applicable methods [13, 12] is to show that a given operad has a quadratic Gröbner basis (as a shuffle operad); this provides a sufficient (but not necessary) condition for Koszulness of an operad. If an operad is Koszul, it necessarily is quadratic, that is has weight-homogeneous relations of weight $2$.

The operads that serve as “building blocks” for operads considered throughout the paper are mostly well known: $\operatorname{Com}$ (commutative associative algebras), $\operatorname{Lie}$ (Lie algebras), $\operatorname{As}$ (associative algebras), $\operatorname{Leib}$ (Leibniz algebras [26]), $\operatorname{Zinb}$ (Zinbiel algebras [27]), $\operatorname{Perm}$ ([associative] permutative algebras [8]), NAP (nonassociative permutative algebras [24], closely related to “right-commutative magma” [15]). All these operads are Koszul, and have a quadratic Gröbner basis.

As we said before, some of our constructions exist both in a “geometric” and a “linear” setting, and are related to each other via the homology functor (which assigns to a topological space $Y$ the graded cocommutative coalgebra $H_{*}(Y)$). To make additional structures transfer easily, we use basic concepts of the theory of polynomial functors. A polynomial functor is a notion that categorifies the notion of a polynomial, and more generally of a formal power series. Polynomial functors provide a useful uniform language to deal with categorical constructions that have “a polynomial flavour”, e.g. when computing sums and products in appropriate categories over specified sets indexing summands/factors in a way that keeps track of the intrinsic structure of the indexing sets.

In precise words, a diagram of sets and set maps

gives rise to a polynomial functor $F:\operatorname{Set}/I\to\operatorname{Set}/J$ defined by the formula

Here _∗ and _! denote, respectively, the right adjoint and the left adjoint of the pullback functor ^∗. More explicitly, the functor is given by

where the last set is considered to be over $J$ via $t_{!}$. Here one can replace $\operatorname{Set}$ by another category where all the appropriate notions make sense. For our purposes, it is enough to consider the case $I=J=*$, in which case the corresponding functors were referred to as polynomial functors in [31], and are called polynomial functors in one variable in more recent literature.

For a systematic introduction to polynomial functors, we refer the reader to the paper [23] and the notes [22] that reflect the state-of-art of the theory.

Let $Y$ be a set and let $\textrm{{NAP}}_{Y}(n)$ be the set of $Y$-trees with vertex set $\left[n\right]={\left\{1,\ldots,n\right\}}$. When $Y$ is a singleton set this is just the set of rooted trees which we denote $\textrm{{RT}}(n)$. The symmetric group $\mathbb{S}_{n}$ acts on $\textrm{{NAP}}_{Y}(n)$ by permuting elements of the vertex set. For a given rooted tree the set of $Y$-labellings is equal to $\operatorname{Hom}(E,Y)=Y^{E}$. Since the number of edges of a tree on ${\left\{n\right\}}$ is always $n-1$, the set of $Y$-labellings is in turn isomorphic to $Y^{n-1}$. Hence

In this way if $Y$ is a topological space then we may also apply a topology to $\textrm{{NAP}}_{Y}(n)$ using the product topology on $Y^{n-1}$.

Now let $T_{1}\in\textrm{{NAP}}_{Y}(n)$ and $T_{2}\in\textrm{{NAP}}_{Y}(m)$ and $i\in\left[n\right]$. We may define a composition $T_{1}\circ_{i}T_{2}\in\textrm{{NAP}}_{Y}(n+m-1)$ by first identifying the root of $T_{2}$ with the vertex $i$ in $T_{1}$. This is a tree and may be rooted by taking the root of $T_{1}$. The edge set is equal to the union $E(T_{1})\amalg E(T_{2})$ of the edge sets of $T_{1}$ and $T_{2}$ and so one inherits an edge labelling by elements of $Y$. It has the vertex set

We then relabel the vertices by elements of $\left[n+m-1\right]$ using the isomorphism which fixes ${\left\{1,\ldots,i-1\right\}}$, shifts the set ${\left\{1,\ldots,m\right\}}$ to ${\left\{i,\ldots,m+i-1\right\}}$ and shifts ${\left\{i+1,\ldots,n\right\}}$ to ${\left\{m+i,\ldots,m+n-1\right\}}$. This gives a rooted $Y$-tree on the vertex set ${\left\{1,\ldots,n+m-1\right\}}$ and so an element $T_{1}\circ_{i}T_{2}\in\textrm{{NAP}}_{Y}(n+m-1)$.

The above theorem gives quadratic relations in the binary generators, the Corollary 6.7 will show that these suffice to present the operad.

Let us finish this section with a few words on $\textrm{{NAP}}_{Y}$-algebras. One convenient way to think of them is via the “right regular module”, since the defining relations say that all the right multiplications

commute with each other. Somewhat more precisely, let $A$ be an object in a symmetric monoidal category $\mathcal{C}$, and let

be a map whose image is an abelian submonoid. Then $A$ is a $\textrm{{NAP}}_{Y}$-algebra enriched in $\mathcal{C}$ with the structure maps given by

This way to approach $\textrm{{NAP}}_{Y}$-algebras gives a source of examples based on $\operatorname{Perm}$-algebras with a family of maps as follows.

One more observation we want to mention in this section is that the construction of the free NAP-algebra mentioned in [24] admits an immediate generalisation to the case of $\textrm{{NAP}}_{Y}$-algebras: the free $\textrm{{NAP}}_{Y}$-algebra enriched in $\operatorname{Set}$ with the generating set $V$ admits a realisation as the set of $Y$-trees whose vertices carry labels from $V$, with the product defined in the same way as we defined the composition in the operad:

In this composition the root of $b$ is joined to the root of $a$ by an edge labelled by $y$; the new root is taken to be the root of $a$.

Let $V$ be a set and $\mathbf{Y}$ be a $V$-tuple of pointed spaces. Let $T$ be a $\mathbf{Y}$-tree with root $r\in V$ and suppose that $\overrightarrow{ij}$ is an edge of $T$ where $i\neq r$. Suppose further that $\overrightarrow{ij}$ is labelled by the basepoint $\bullet\in Y_{i}$. Then we say that $\overrightarrow{ij}$ is a reducible edge and that $T$ is reducible. Since $i$ is not the root there is a unique incoming edge $\overrightarrow{ki}$ which is labelled by some $y\in Y_{k}$. We define $T_{ij}$ to be the $\mathbf{Y}$-tree given by removing the edge $\overrightarrow{ij}$ and adding the edge $\overrightarrow{kj}$ with the label $y\in Y_{k}$. We say that $T_{ij}$ is a reduction of $T$.

In the spirit of how we approached $\textrm{{NAP}}_{Y}$-algebras, a $\textrm{{BCact}}_{Y}$-algebra enriched in a symmetric monoidal category $\mathcal{C}$ is a $\textrm{{NAP}}_{Y}$-algebra enriched in $\mathcal{C}$ where the operation is associative, and

Let $Y$ be a topological space and let $P$ be a subset of $Y$. We define the fundamental groupoid $\pi_{1}(Y,P)$ to be the groupoid with objects the points $p\in P$ and morphisms the homotopy classes of paths in $Y$ which start and end in elements of $P$. The composition is by concatenation of paths and the units are supplied by the constant paths. So if $(Y,\bullet)$ is a pointed space then $\pi_{1}(Y,{\left\{\bullet\right\}})$ is the fundamental group of $Y$. Let $(Y,\bullet)$ be a pointed space and let $P\in Y$ be a set of points which contains $\bullet$ and such that each path connected component of $Y$ contains a single point of $P$. This may be seen as a section of the map

Then by the functoriality of $\textrm{{BCact}}_{(-)}$ there is a pair of operad maps

which serves to pick out a single element in each path connected component of $\textrm{{BCact}}_{Y}$. The fundamental groupoid functor preserves products and colimits and so $\pi_{1}(\textrm{{BCact}}_{Y};\textrm{{BCact}}_{P})$ is an operad in the category of groupoids.

From now on we will restrict $Y$ to be a path connected space, so $P={\left\{\bullet\right\}}$. In this case $\textrm{{BCact}}_{P}\cong\operatorname{Perm}$, the operad for permutative algebras — each of the $n$ elements is given by a corolla. So we see that $\textrm{{BCact}}_{Y}(n)$ is made up of $n$ components and the action of $\mathbb{S}_{n}$ gives isomorphisms between them. Denote by $\textrm{{BCact}}_{Y}(n)_{r}$ the component consisting of trees with root $r$.

We have already seen that $\pi_{1}(\textrm{{BCact}}_{Y},\textrm{{BCact}}_{P})$ is an operad, to give the composition maps we need only describe the compositions on the generating morphisms. In fact since we have $g\circ_{i}h=(g\circ_{i}e).(e\circ_{i}h)$ we need only describe the compositions of generators with identity maps.