Permutads via operadic categories, and the hidden associahedron

TL;DR

This paper explores the structure of permutads within operadic categories, demonstrating their relation to cellular chains of permutohedra, and establishes their Koszul properties and minimal models, advancing the understanding of algebraic and topological operad theory.

Contribution

It proves that cellular chains of permutohedra form the minimal model of the terminal permutad and shows the Koszul self-duality of the terminal Per-operad, introducing new perspectives on permutads and operadic categories.

Findings

Cellular chains of permutohedra form the minimal model of the terminal permutad.

The terminal Per-operad is Koszul self-dual.

Strongly homotopy permutads are described as algebras of the minimal model.

Abstract

The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco that the cellular chains of the permutohedra form the minimal model of the terminal permutad which is moreover, in the sense we define, self-dual and Koszul. In the second part we study Koszulity of Per-operads. Among other things we prove that the terminal Per-operad is Koszul self-dual. We then describe strongly homotopy permutads as algebras of its minimal model. Our paper shall advertise analogous future results valid in general operadic categories, and the prominent role of operadic (op)fibrations in the related theory.

Equations

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$r^{-1}(i)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A=\{A(\underline{n})\in \mathtt{Vec}|\underline{n}\in \mathtt{Fin}\},$$

$$(A\odot B)(\underline{n}):=\bigoplus_{r\in{\tt Surj}(\underline{n},\underline{2})}\big{\{}A({r}^{-1}(1))\otimes A({r}^{-1}(2))\big{\}}.$$

$$\upsilon (n_1+\cdots +n_{j-1}+1)<\upsilon (n_1+\cdots +n_{j-1}+2)<\cdots <\upsilon (n_1+\cdots +n_j).$$

$$(1,2,3,4,5,6,7,8)⟼(2,4,1,3,6,5,7,8)\in {\Sigma }_8.$$

$$\mathtt{Surj}(\underline{n},\underline{k})\cong \underset{n_1+\cdots +n_k=n}{⨆}\mathrm{Sh}(n_1,\dots ,n_k)$$

$$I_j:=\{i|r(i)=j\},1\le j\le k.$$

$$r_j=\mathrm{card}\{i|r(i)=j\},and\epsilon (r):=\mathrm{signum}({\upsilon }_r)\in \{-1,+1\},1\le j\le k,$$

$${\pi }_{(1,2+3)}:\mathrm{Sh}(n_1,n_2,n_3)⟶\mathrm{Sh}(n_1,n_2+n_3)$$

$${\pi }_{(1+2,3)}:\mathrm{Sh}(n_1,n_2,n_3)⟶\mathrm{Sh}(n_1+n_2,n_3)$$

$${\pi }_{(1,2)}:\mathrm{Sh}(n_1,n_2,n_3)⟶\mathrm{Sh}(n_1,n_2)$$

$${\pi }_{(2,3)}:\mathrm{Sh}(n_1,n_2,n_3)⟶\mathrm{Sh}(n_2,n_3)$$

$${\pi }_{(1,2+3)}([2,4|1,3,6|5,7,8])$$

$${\pi }_{(1+2,3)}([2,4|1,3,6|5,7,8])$$

$${\pi }_{(1,2)}([2,4|1,3,6|5,7,8])$$

$${\pi }_{(2,3)}([2,4|1,3,6|5,7,8])$$

$$\mathrm{Sh}(n_1,n_2,n_3)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\bullet }_{\tau }:A({\underline{n}}_1)\otimes A({\underline{n}}_2)⟶A(\underline{n})$$

$${\bullet }_{\gamma }(11\otimes {\bullet }_{\delta })={\bullet }_{\lambda }({\bullet }_{\sigma }\otimes 11)$$

$$a{\bullet }_{\gamma }(b{\bullet }_{\delta }c)=(a{\bullet }_{\sigma }b){\bullet }_{\lambda }c,$$

$$(11\times \delta )\cdot \gamma :=[{\pi }_{(2,3)}\times {\pi }_{(1,2+3)}{]}^{-1}(\delta \times \gamma ),(\sigma \times 11)\cdot \lambda :=[{\pi }_{(1,2)}\times {\pi }_{(1+2,3)}{]}^{-1}(\sigma \times \lambda ).$$

$$A(\underline{n}):=A_n,n\ge 1,and{\bullet }_{\tau }:=\bullet for\; each\; shuffle\tau ,$$

$$t^{-1}(1)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\{g_1,\dots ,g_a\}=\{j|t(j)=2\}\subset \{1,\dots ,m\},$$

$$t(11\times s)(j):=\{\begin{array}{ll}1,& ifj\notin \{g_1,\dots ,g_a\},\\ 2,& ifj=g_{k}for\; somekwiths(k)=1,\; and\\ 3,& ifj=g_{k}for\; somekwiths(k)=2.\end{array}$$

$$\{h_1,\dots ,h_b\}=\{j|u(j)=1\}\subset \{1,\dots ,m\},$$

$$u(v\times 11)(j):=\{\begin{array}{ll}1,& ifj=h_{k}for\; somekwithv(k)=1.\\ 2,& ifj=h_{k}for\; somekwithv(k)=2,\; and\\ 3,& ifj\notin \{h_1,\dots ,h_b\}.\end{array}$$

$${♢}_r:A(r^{-1}(1))\otimes A(r^{-1}(2))⟶A(\underline{n})$$

$${♢}_u({♢}_v\otimes 11)={♢}_t(11\otimes {♢}_s)$$

$$t(11\times s)=u(v\times 11).$$

$${\mathbb{P}}^k(B)(\underline{n}):=\underset{r\in \mathtt{Surj}(\underline{n},\underline{k})}{⨁}\underset{i\in \underline{k}}{⨂}B(r^{-1}(i)).$$

Full text

Supported by Praemium Academiae, grant GA ČR 18-07776S and RVO: 67985840.

1. I Classical approach to permutads

2. 1 Permutads – recollections

3. 2 Koszul duals of permutads

4. 3 Koszulity of permutads

5. II Operadic category approach

6. 4 Permutads as algebras over the terminal operad

7. 5 Free ${\tt Per}$-operads

8. 6 Minimal model of the terminal ${\tt Per}$-operad and homotopy permutads

9. 7 Koszul duals of ${\tt Per}$-operads

10. 8 Koszulity of ${\tt Per}$-operads

Motivations and the main results

Recall that differential graded (abbreviated dg) associative algebras are triples $A=(A,\bullet,d)$ consisting of a graded vector space $A$, an associative degree-[math] product $\bullet:A\otimes A\to A$, and a differential $d$ which is a derivation with respect to the product. Associative algebras are however too rigid for some applications in homological algebra and deformation theory. For instance, having an algebra $A=(A,\bullet,d)$ as above and a dg vector space $(A^{\prime},d^{\prime})$ chain homotopy equivalent to $(A,d)$, there is in general no associative multiplication on $(A^{\prime},d^{\prime})$ induced by the one on $(A,d)$.

The way around is to embed dg associative algebras into the category of strongly homotopy associative algebras. These objects, also called $A_{\infty}$-algebras, are dg vector spaces $(A,d)$ together with operations $\{\mu_{k}\mskip-3.0mu:\mskip-3.0muA^{\otimes k}\to A\}_{k\geq 2}$ such that $\mu_{2}\mskip-3.0mu:\mskip-3.0muA\otimes\mskip-3.0muA\to A$ is associative up to the chain homotopy $\mu_{3}$, and ${\mu_{2},\ldots,\mu_{k-1}}$ satisfy, for each $k\geq 4$, a specific coherence relation up to the chain homotopy $\mu_{k}$, cf. [19, Example II.3.132] for a precise definition. $A_{\infty}$-algebras indeed behave well with respect to chain homotopy equivalences of their underlying dg vector spaces [17].

Operad theory explains why it is so. In the operadic context, dg associative algebras emerge as algebras for the non-$\Sigma$ (non-symmetric) operad $\underline{{\mathcal{A}}{\it ss}}$. It is a quadratic operad, and its quadratic dual $\underline{{\mathcal{A}}{\it ss}}^{!}$ is isomorphic to $\underline{{\mathcal{A}}{\it ss}}$ [8, Thm. 2.1.11].111The results and definitions cited here were formulated for $\Sigma$- (symmetric) operads, but their obvious non-$\Sigma$ versions hold as well. Definition 4.2.14 of [8] thus interprets $A_{\infty}$-algebras as algebras for the dual dg operad ${\mathbb{D}}(\underline{{\mathcal{A}}{\it ss}})$, the dg operad defined as the operadic bar construction applied to the suspended piece-wise linear dual of $\underline{{\mathcal{A}}{\it ss}}$ [8, page 245]. The dual dg operad is, by construction, free with a quadratic differential. In particular, it is cofibrant, and its cofibrancy guarantees good homotopy properties of its algebras, which is a principle latent in topology since Boardman-Vogt’s [5], spelled out for algebra e.g. in [4, 15].

Strongly homotopy algebras are algebraic cousins of $A_{\infty}$-spaces, which are topological spaces with an action of the operad ${\underline{\mathcal{K}}}$ of Stasheff’s associahedra [20]. The operad ${\underline{\mathcal{K}}}$ lives in the category of contractible cell complexes; its arity-two piece ${\underline{\mathcal{K}}}(2)$ is the point, ${\underline{\mathcal{K}}}(3)$ is the interval and ${\underline{\mathcal{K}}}(4)$ the pentagon. A portrait of ${\underline{\mathcal{K}}}(5)$ can be found in [19, page 10].

A bridge between topology and algebra is an explicit isomorphism between the cellular chain complex $\hbox{\it CC}_{*}({\underline{\mathcal{K}}})$ of Stasheff’s operad and the dual dg operad ${\mathbb{D}}(\underline{{\mathcal{A}}{\it ss}})$, constructed in [16, Example 4.8]. The contractibility of the pieces of ${\underline{\mathcal{K}}}$ together with the isomorphism ${\mathbb{D}}(\underline{{\mathcal{A}}{\it ss}})\cong\hbox{\it CC}_{*}({\underline{\mathcal{K}}})$ implies that the dg vector space ${\mathbb{D}}(\underline{{\mathcal{A}}{\it ss}})(n)$ is acyclic in positive dimensions for each arity $n$. One can easily derive from this that the canonical map ${\mathbb{D}}(\underline{{\mathcal{A}}{\it ss}})\to\underline{{\mathcal{A}}{\it ss}}$ is a piece-wise homology isomorphism, therefore ${\mathbb{D}}(\underline{{\mathcal{A}}{\it ss}})$ is the minimal model of $\underline{{\mathcal{A}}{\it ss}}$ [19, Def. II.3.124]. This, by [16, Prop. 2.6], implies that $\underline{{\mathcal{A}}{\it ss}}$ is Koszul.

Let us point out that Koszulity is an important property of quadratic operads that e.g. implies the existence of a well-behaved deformation theory for the related category of algebras. The rôle of the contractibility of Stasheff’s associahedra in this context was noticed already in the author’s 1994 article [14].

Another distinguishing feature of $\underline{{\mathcal{A}}{\it ss}}$ is that it is the linearization of the terminal non-$\Sigma$ operad in the category of sets, i.e. of the ${\tt Set}$-operad whose all pieces equal the one-point set $*$ and all structure operations are the identifications $*\times\cdots\times*=*$. Abusing slightly the terminology, we will call $\underline{{\mathcal{A}}{\it ss}}$ the terminal non-$\Sigma$ operad as well. The above observations can be then concisely formulated as

Fact**.**

The cellular chain complex of the Stasheff’s associahedron is the minimal model of the terminal non-$\Sigma$-operad. This terminal operad is quadratic Koszul.

The original humble aim of this note was to formulate and prove an analog of this Fact for permutads, using indications provided by J.-L. Loday and M. Ronco in [10]. We however decided to extend it to an advertisement for the theory of operadic categories developed in [2] and [3], using permutads as an excuse.

Part 1, independent of the theory of operadic categories, starts by recalling, following [10], permutads and the related notions. We then introduce quadratic permutads and their Koszul duals and formulate the Koszulity property for quadratic permutads in terms of a suitably defined dual bar construction. We close Part 1 by a permutadic analog of the Ginzburg-Kapranov power series test [8, Theorem 3.3.2]. The results of the first part, namely Proposition 19 and Theorem 22 combined with [10, Proposition 5.4], give the following wished-for permutadic analog of the Fact above:

Theorem A**.**

The cellular chain complex of the permutohedron222Sundry definitions of the permutohedron are assembled in Appendix 2 to [10]. is the minimal model of the terminal permutad. This terminal permutad is quadratic Koszul.

An important feature of the dual bar construction333I.e.  the bar construction applied to the linear dual. ${\mathbb{D}}(A)$ of a permutad $A$ introduced in Definition 18 is that it is again a permutad. This self-duality is not automatic. For instance, the dual bar construction of a commutative associative algebra is a Lie algebra, the dual bar construction of a modular operad is a twisted modular operad, &c.

An explanation of self-duality of permutads is offered by [7, Definition 5] which presents them as associative algebras in the category of (graded) vector spaces with an unusual symmetric monoidal structure (2), while the self-duality of associative algebras is classical. We however give an alternative explanation that uses the theory of operadic categories, which are the subject of

Part 2. As shown in [2, §14.4], there exists an operadic category ${\tt Per}$ such that permutads are algebras, in the sense of [3, Definition 1.20], over the terminal ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$; we formulate this statement with a slightly different proof as Proposition 26. The self-duality of the category of permutads follows from Theorem 14.4 of [2] which says that the ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$ governing permutads is binary quadratic and self-dual. We will analyze this phenomenon in the context of general operadic categories in our future work. We complement these results by proving:

Theorem B**.**

The terminal ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$ is Koszul.

As in the case of ‘classical’ operads, Koszulity of ${\sf 1}_{{\tt Per}}$ leads to an effective description of its minimal model ${\mathfrak{M}}$ which we give in the proof of Theorem 32. Algebras over ${\mathfrak{M}}$ are, due to the philosophy pioneered in [16], strongly homotopy permutads. Their explicit description in terms of operations and axioms is given in Proposition 35.

Where the associahedron hides? The present article shall also illustrate the potential of Grothendieck’s construction in operadic categories and the related discrete opfibrations [2, Subsect. 5.2]. Let ${\mathbb{\Delta}_{\rm semi}}$ be the operadic category of finite ordinals and their order-preserving surjections; recall that ${\mathbb{\Delta}_{\rm semi}}$-operads are ordinary constant-free non-$\Sigma$-operads. It turns out that the operadic category ${\tt Per}$ is the Grothendieck’s construction applied to a certain ${\mathbb{\Delta}_{\rm semi}}$-cooperad ${\mathcal{C}}$. One thus has a discrete opfibration ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$.

A consequence is that the restrictions along ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$ of free non-$\Sigma$-operads are free ${\tt Per}$-operads. In particular, the restriction ${\rm des}^{*}({\underline{\mathcal{K}}})$ of Stasheff’s operad of the associahedra turns out to be the convex polyhedral realization of the minimal model ${\mathfrak{M}}$ of the terminal ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$, cf. Remark 38.

In Theorem 48 we prove that the restriction along ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$ brings Koszul non-$\Sigma$ operads into Koszul ${\tt Per}$-operads. This is a particular case of an important feature of opfibrations between operadic categories, and an advertisement for our future work.

Conventions. Our background monoidal category will be the category ${\tt Vec}$ of differential graded, or dg for short, vector spaces $V=\bigoplus_{k\in{\mathbb{Z}}}V_{k}$ over a field ${\mathbb{k}}$ of characteristic [math]; the preferred degree of differentials will be $-1$. The linear duals are taken degree-wise, i.e. the degree $k$ component of the dual $V^{*}$ of a graded space above will be ${\tt Vec}(V_{k},{\mathbb{k}})$, $k\in{\mathbb{Z}}$.

If not stated otherwise, all algebra-like objects (monoidal categories, permutads) will be nonunital. Operadic categories, their operads and algebras over these operads were introduced in [3, §I.1]. The standard reference for ‘classical’ operads, quadratic duality and Koszulness is [8] or more recent [11] or [19].

The cobar construction of a linear dual of an algebra or permutad will be called the dual bar construction, while the cobar construction applied to the linear dual of an operad will be called, following [8], the dual dg operad.

Acknowledgment. I wish to express my gratitude to Vladimir Dotsenko for presenting permutads in a broader context to me and turning my attention to [7]. Also suggestions of the referee lead to a substantial improvement of the paper.

Part I Classical approach to permutads

1. Permutads – recollections

Permutads appear under various names in various disguises. Following [7] and [10] we recall some of them in this section. The most concise definition uses the skeletal category ${\tt Fin}$ of finite non-empty sets with objects the ordinals $\underline{n}:=\{{1,\ldots,n}\}$, $n\geq 1$, and its subcategory ${\tt Surj}$ of surjections. For any order-preserving surjection $f:A\to B$ of finite ordered sets there exists a unique $\alpha:\underline{n}\to\underline{k}\in{\tt Surj}$ in the commutative diagram

[TABLE]

where the vertical arrows are order-preserving isomorphisms. In this sense, every order-preserving surjection $f:A\to B$ of finite ordered sets may be interpreted as a morphism in ${\tt Surj}$. For r:\underline{m}{\hbox{ \twoheadrightarrow }}\underline{n}\in{\tt Surj} and $i\in\underline{n}$ we denote by ${r}^{-1}(i)$ the pullback

[TABLE]

in ${\tt Fin}$. Notice that ${r}^{-1}(i)=\underline{c}$, with $c\geq 1$ the cardinality of the set-theoretic preimage of $i$ under $r$. We emphasize that ${r}^{-1}(i)$ does not denote the set-theoretic inverse image, but the categorial pull-back, and serves as an example of a fiber in an operadic category. Permutads live in the category ${\tt Coll}$ of collections

[TABLE]

which we consider as a (non-unital) monoidal category with the shuffle product [7, Definition 1]

[TABLE]

The coproducts of graded vector spaces obviously commute with $\odot$ from both sides, which is the property that, according to [1], guarantees that the monoidal category ${\tt Coll}$ behaves in most aspects as the standard monoidal category of graded vector spaces. The following definition is taken from [7].

Definition 1**.**

A permutad, also called a shuffle algebra, is a monoid for the monoidal product (2).

Remark 2**.**

The term shuffle algebra comes from the fact that surjections $r$ in (2) are in one-to-one correspondence with $(n_{1},n_{2})$-shuffles for some $n_{1}\mskip-3.0mu+\mskip-3.0mun_{2}=n$. Let us recall what shuffles are. Given natural numbers $n_{1},\dots,n_{k}\geq 1$ such that $n_{1}\mskip-3.0mu+\mskip-3.0mu\cdots\mskip-3.0mu+\mskip-3.0mun_{k}=n$, an $(n_{1},\dots,n_{k})$*-shuffle * is a permutation $\upsilon\in\Sigma_{n}$ such that for any $j={1,\ldots,k}$ one has

[TABLE]

It is clear that $(n_{1},\dots,n_{k})$-shuffles are in one-to-one correspondence with ordered partitions $[I_{1}|\cdots|I_{k}]$ of $\underline{n}=\{{1,\ldots,n}\}$ into $k$ disjoint nonempty subsets, with $I_{j}$ being, for $1\leq j\leq k$, the subset of $\underline{n}$ consisting of the numbers in (3). When convenient, we use partitions to denote shuffles. Thus, for instance, $[2,4|1,3,6|5,7,8]$ denotes the $(2,3,3)$-shuffle

[TABLE]

Denoting by ${\rm Sh}(n_{1},\dots,n_{k})\subset\Sigma_{n}$ the set of all $(n_{1},\dots,n_{k})$-shuffles, one has the isomorphism

[TABLE]

where $\bigsqcup$ denotes the disjoint union, which identifies $r\in{\tt Surj}(\underline{n},\underline{k})$ with the partition $[I_{1}|\cdots|I_{k}]$ given by

[TABLE]

For the future use we denote

[TABLE]

where $\upsilon_{r}\in\Sigma_{n}$ is the shuffle corresponding to $r$ in the correspondence (4).

We are going to give two alternative, explicit definitions of permutads. The first one emphasizes their combinatorial nature, the second one is tailored for the purposes of our paper. Let us introduce necessary auxiliary notation. We will need

[TABLE]

given by $\pi_{(1,2+3)}([I_{1}|I_{2}|I_{3}]):=[I_{1}|I_{2}\cup I_{3}]$, and the map

[TABLE]

given by the obvious similar formula. We also define the map

[TABLE]

by $\pi_{(1,2)}([I_{1}|I_{2}|I_{3}]):=[{\rm st}(I_{1})|{\rm st}(I_{2})]$, where the standardization ${\rm st}:I_{1}\cup I_{2}\to\{1,\ldots,n_{1}\mskip-3.0mu+\mskip-3.0mun_{2}\}$ is the unique monotonous map. The map

[TABLE]

is likewise given by $\pi_{(2,3)}([I_{1}|I_{2}|I_{3}]):=[{\rm st}(I_{2})|{\rm st}(I_{3})]$. For example

[TABLE]

Notice that the above maps induce isomorphisms in the diagram

[TABLE]

We are ready to give the following explicit

Definition 3**.**

A permutad is a collection $A\in{\tt Coll}$ together with operations

[TABLE]

given for each shuffle $\tau\in{\rm Sh}(n_{1},n_{2})$ with $n_{1},n_{1}\geq 1$, $n=n_{1}\mskip-3.0mu+\mskip-3.0mun_{2}$. These operations are required to satisfy

[TABLE]

for each $\hbox{{\cyr Sh}}\in{\rm Sh}(n_{1},n_{2},n_{3})$, $n_{1},n_{2},n_{3}\geq 1$, with $\gamma=\pi_{(1,2+3)}(\hbox{{\cyr Sh}})$, $\delta=\pi_{(2,3)}(\hbox{{\cyr Sh}})$, $\lambda=\pi_{(1+2,3)}(\hbox{{\cyr Sh}})$ and $\sigma=\pi_{(1,2)}(\hbox{{\cyr Sh}})$.

In elements, equation (Sh) can of course be written as

[TABLE]

for $a\in A({\underline{n}}_{\,1})$, $b\in A({\underline{n}}_{\,2})$ and $c\in A({\underline{n}}_{\,3})$. To rewrite Definition 3 into the form closer to [10], we put, using the isomorphisms in (6),

[TABLE]

Shuffle algebras then appear as structures with operations (7) that satisfy (Sh) for each $\delta,\gamma,\sigma,\lambda$ such that $(1\mskip-3.0mu\mskip-3.0mu1\times\delta)\cdot\gamma=(\sigma\times 1\mskip-3.0mu\mskip-3.0mu1)\cdot\lambda$ in ${\rm Sh}(n_{1},n_{2},n_{3})$. This is precisely the definition given in [10, Subsect. 3.4].

Example 4**.**

Assume that $A=\bigoplus_{n\geq 1}A_{n}$ is a graded associative algebra with a product $\bullet:A\otimes A\to A$. If we define

[TABLE]

then (7) follows from the associativity of the product. Thus graded associative algebras provide examples of permutads. Other examples can be found in [7, Subsect. 2.2] or in the second half of [10].

Let us translate Definition 3 into the language of surjections. For surjections $t:\underline{m}\to\underline{2}$ and $s:{t}^{-1}(2)\to\underline{2}$ we denote by $t(1\mskip-3.0mu\mskip-3.0mu1\times s):\underline{m}\to\underline{3}$ the surjection defined by the commutativity of the diagrams

[TABLE]

where $\iota_{1}(1):=1$, $\iota_{2,3}(1):=2$ and $\iota_{2,3}(2):=3$. More explicit, though less conceptual, description of $t(1\mskip-3.0mu\mskip-3.0mu1\times s)\in{\tt Surj}(\underline{m},\underline{3})$, goes as follows. For $t:\{{1,\ldots,m}\}\to\{1,2\}$, let $g_{1}<\cdots<g_{a}$ be numbers such that

[TABLE]

so that $t^{-1}(2)=\underline{a}$ and $s:\{{1,\ldots,a}\}\to\{1,2\}$. Then $t(1\mskip-3.0mu\mskip-3.0mu1\times s):\{{1,\ldots,m}\}\to\{1,2,3\}$ is given by

[TABLE]

Analogously we define, for $u\in{\tt Surj}(\underline{m},\underline{2})$ and $v:{u}^{-1}(1)\to\underline{2}$, the surjection $u(v\times 1\mskip-3.0mu\mskip-3.0mu1)\in{\tt Surj}(\underline{m},\underline{3})$ as follows. Let $h_{1}<\cdots<h_{b}$ be such that

[TABLE]

so $v:\{{1,\ldots,b}\}\to\{1,2\}$. Then

[TABLE]

We leave as an exercise to express $u(v\times 1\mskip-3.0mu\mskip-3.0mu1)$ in the diagrammatic language of (9).

If the surjection $t$ above corresponds, under isomorphism (4), to a shuffle $\gamma$, $s$ corresponds to $\delta$, $u$ to $\lambda$ and $v$ to $\sigma$, then $t(1\mskip-3.0mu\mskip-3.0mu1\times s)$ corresponds to $(1\mskip-3.0mu\mskip-3.0mu1\times\delta)\cdot\gamma$ in (8), and $u(v\times 1\mskip-3.0mu\mskip-3.0mu1)$ to $(\sigma\times 1\mskip-3.0mu\mskip-3.0mu1)\cdot\lambda$. With these definitions at hand we may formulate the following

Definition 5**.**

A permutad is a collection $A\in{\tt Coll}$ together with operations

[TABLE]

defined for each surjection $r\in{\tt Surj}(\underline{n},\underline{2})$. These operations shall satisfy

[TABLE]

for each $m\geq 1$ and surjections $u,t:\underline{m}\to\underline{2}$, $s:{t}^{-1}(1)\to\underline{2}$ and $v:{u}^{-1}(2)\to\underline{2}$ such that

[TABLE]

Definitions 3 and 5 suggest that permutads are close in nature to operads. One thus sometimes uses, as in [10], a shifted grading $A_{n+1}:=A(\underline{n})$, $n\geq 1$. The underlying collection $A_{2},A_{3},\ldots$ of a permutad is then the same as the underlying collection of a $1$-connected non-unital non-symmetric (non-$\Sigma$) operad.

Let us recall from [10] a monadic definition of permutads. For a collection $B\in{\tt Coll}$, define ${\mathbb{P}}(B)\in{\tt Coll}$ by ${\mathbb{P}}(B):=\bigoplus_{k\geq 1}{\mathbb{P}}^{k}(B)$, where

[TABLE]

Remark 6**.**

Assume that $A$ is a permutad as in Definition 5. Since clearly ${\mathbb{P}}^{1}(A)\cong A$ and

[TABLE]

structure operations (10) assemble into one map

[TABLE]

As shown in [10, Section 1.6], there exists a natural morphism of collections

[TABLE]

which, together with the obvious inclusion $\iota_{B}:B\hookrightarrow{\mathbb{P}}(B)$, makes ${\mathbb{P}}(-)$ a monad in ${\tt Coll}$. One has the fourth

Definition 7**.**

A permutad is an algebra for the monad ${\mathbb{P}}:{\tt Coll}\to{\tt Coll}$.

As a consequence of classical statements, see e.g. [12, Theorem VI.2.1], ${\mathbb{P}}(B)$ is the free permutad generated by $B$.

Example 8**.**

444Quoting a Czech physicist: “Old men give only good examples, since they are unable to give bad ones.”

An important rôle will be featured by the permutad $\it per\hskip-1.00006pt{\mathcal{A}}s$ with \hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}(\underline{n}):={\mathbb{k}} for all $\underline{n}\in{\tt Fin}$, with all operations (7) the canonical isomorphisms ${\mathbb{k}}\otimes{\mathbb{k}}\cong{\mathbb{k}}$. It is the linearization of the terminal permutad in the category ${\tt Set}$ of sets, so we will call it, being aware of slight abuse of terminology, the terminal permutad.

Example 9**.**

The permutohedron $P_{n+1}$ is, for $n\geq 1$, a convex polytope whose faces are labelled by planar rooted trees with levels, see e.g. [21] or [10, Appendix 2]. Its cellular chain complex

[TABLE]

is a permutad in the category of dg vector spaces whose underlying permutad is free, generated by the collection $B$ which has

[TABLE]

the ground field ${\mathbb{k}}$ placed in degree $n\mskip-3.0mu-\mskip-3.0mu1$, cf. [10, Theorem 5.3]. In (15), $c_{n-1}:=\uparrow^{n-1}\mskip-3.0mu1$, $1\in{\mathbb{k}}$, is the generator of $\uparrow^{n-1}{\mathbb{k}}$. Indeed, formula (12) gives

[TABLE]

where $r_{i}$ is the cardinality of the set-theoretic preimage of $i\in\underline{k}$ via r:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}. The set ${\tt Surj}(\underline{n},\underline{k})$ is isomorphic to the set of ordered decompositions of $\{{1,\ldots,n}\}$ into $k$ disjoint non-empty subsets which is, in turn, isomorphic to the set of planar rooted trees with $n\mskip-3.0mu+\mskip-3.0mu1$ leaves and $k$ levels, see e.g. [13, §1.3].

Explicitly, the isomorphism $\hbox{\it CC}_{*}(P)\cong{\mathbb{P}}(B)$ identifies the ‘big’ $n$-dimensional cell of $P_{n}$ with the generator $c_{n}$ of ${\mathbb{P}}(B)$, $n\geq 0$. A formula for the boundary operator of $\hbox{\it CC}_{*}(P)$ written in terms of ${\mathbb{P}}(B)$ can be found in [10, §9.3]. We will return to the permutohedron in Proposition 19.

Example 10**.**

Each permutad $A$ admits a permutadic suspension ${\mathfrak{s}}A$. Its underlying collection is {\mathfrak{s}}A(\underline{n}):=\hbox{\uparrow^{n}\hskip-3.00003ptA(\underline{n})}, $n\geq 1$, and the structure operations (10)

[TABLE]

are defined by

[TABLE]

in which \hbox{\tiny\diamondsuit}_{r} is the structure operation of $A$, $a_{i}\in A({r}^{-1}(i))$, $i=1,2$, and $r_{1},r_{2}$ and $\varepsilon(r)$ are as in (5).

Remark 11**.**

There exist symmetric versions of permutads, called twisted associative algebras, whose underlying collections bear actions of the symmetric groups, see [6, Chapter 4] for their definition. Koszul duality for these objects is treated in a recent preprint [9]. We believe that the methods developed in the present article can be easily modified also to twisted associative algebras.

2. Koszul duals of permutads

Most of the notions in this section and in the first half of Section 3 are straightforward translations of the analogous standard notions for associative algebras given by replacing the usual ambient monoidal structure of vector spaces by the shuffle product (2). Our preferred form of structure operations of permutads will be that in Definition 5 based on surjections. Let us start by recalling the following definition of [10, §4.4].

Definition 12**.**

A permutad $A$ is quadratic if it is of the form ${\mathbb{P}}(B)/(S)$, where $(S)$ is the permutadic ideal generated by a subspace $S\subset{\mathbb{P}}^{2}(B)$. Such a quadratic permutad $A$ is binary, if $B(\underline{n})=0$ for $n\not=1$.

Example 13**.**

Let us denote, abusing the notation again, by ${\mathbb{P}}(\mu)$ the free permutad generated by one generator $\mu\in{\mathbb{P}}(\mu)(\underline{1})$. As stated in [10, Corollary 4.7], the terminal permutad $\it per\hskip-1.00006pt{\mathcal{A}}s$ recalled in Example 4 is binary quadratic,

[TABLE]

with $S\subset{\mathbb{P}}^{2}(\mu)={\mathbb{P}}(\mu)(\underline{2})$ the ${\mathbb{k}}$-linear span

[TABLE]

where $1\mskip-3.0mu\mskip-3.0mu1:\underline{2}\to\underline{2}$ the identity and $(21):\underline{2}\to\underline{2}$ the transposition $1,2\mapsto 2,1$. The verification of (16) is simple. Formula (12) immediately gives that

[TABLE]

the ${\mathbb{k}}$-linear span of the symmetric group of $n$ elements, while modding out by the ideal $(S)$ identifies any two permutations in $\Sigma_{n}$ that differ by transposition of adjacent elements. Since transpositions act transitively, one gets {\mathbb{P}}(\mu)/(S)(\underline{n})\cong{\mathbb{k}}\cong\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}(\underline{n}), $n\geq 1$, as expected.

Every binary quadratic permutad $A$ as in Definition 12 possess its Koszul dual $A^{!}$, which is a binary quadratic permutad defined as

[TABLE]

where $\uparrow\hskip-3.50006ptB^{*}$ is the suspension of the component-wise linear dual of the generating collection $B$, and $S^{\perp}\subset{\mathbb{P}}^{2}(\uparrow\hskip-3.50006ptB^{*})$ is the annihilator of $S\subset{\mathbb{P}}^{2}(B)$ with respect to the obvious natural degree $-2$ pairing between

[TABLE]

and

[TABLE]

Proposition 14**.**

The terminal permutad $\it per\hskip-1.00006pt{\mathcal{A}}s$ is Koszul self-dual in the sense that its Koszul dual \hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!} is isomorphic to its permutadic suspension {\mathfrak{s}}\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}. Explicitly,

[TABLE]

with the structure operation (10) the composition

[TABLE]

multiplied with $\varepsilon(r)\in\{-1,+1\}$, where $r_{1}$, $r_{2}$ and $\varepsilon(r)$ have the same meaning as in (5).

Proof.

By the definition of the Koszul dual, one has

[TABLE]

where $\mu^{\uparrow}\in{\mathbb{P}}(\mu^{\uparrow})(\underline{1})$ is a degree $1$ generator and $S^{\perp}$ the span

[TABLE]

By formula (12),

[TABLE]

while modding out by the ideal $(S^{\perp})$ introduces the relation $\sigma^{\prime}\sim-\sigma^{\prime\prime}$ for each $\sigma^{\prime}\in\Sigma_{n}$ and $\sigma^{\prime\prime}$ obtained from $\sigma^{\prime}$ by a transposition of two adjacent elements. Therefore the assignment

[TABLE]

leads to an isomorphism $\uparrow^{n}\mskip-3.0mu{\mathbb{k}}\cong{\mathbb{P}}(\mu^{\uparrow})/(S^{\perp})(\underline{n})$. The advertised formula for the structure operations easily follows from the description of the operations in the free permutad ${\mathbb{P}}(\mu^{\uparrow})$ given in [10, Section 1.6]. ∎

3. Koszulity of permutads

We start this section with a permutadic version of the cobar construction and the related dual bar construction. We then establish that the dual bar construction of the Koszul dual \hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!} of the terminal permutad $\it per\hskip-1.00006pt{\mathcal{A}}s$ is isomorphic to the cellular chain complex of the permutohedron. Finally, we prove that $\it per\hskip-1.00006pt{\mathcal{A}}s$ is Koszul and formulate a test for Koszulity of permutads. The first definition is a harmless formal dual of Definition 5.

Definition 15**.**

A copermutad is a collection $C\in{\tt Coll}$ together with operations

[TABLE]

satisfying the obvious dual versions of axioms (11).

Example 16**.**

Assume that $A$ is a permutad such that $A(\underline{n})$ is, for each $n\geq 1$, either finite-dimensional, or non-negatively or non-positively graded vector space of finite type. Then its component-wise linear dual $A^{*}={\{A^{*}(\underline{n})\}_{n\geq 1}}$ is a copermutad. This is in particular satisfied if $A$ is binary quadratic as in Definition 12, with $B(1)$ finite-dimensional.

Definition 17**.**

A degree $s$ derivation of a permutad $A$ is a degree $s$ linear map $\varsigma:A\to A$ of collections such that

[TABLE]

for the structure operations \hbox{\tiny\diamondsuit}_{r} in (10). In elements,

[TABLE]

One easily verifies that each degree $s$ map of collections $\sigma:B\to{\mathbb{P}}(B)$ uniquely extends to a degree $s$ derivation $\varsigma$ of the free permutad ${\mathbb{P}}(B)$ satisfying $\varsigma|_{B}=\sigma$. Let $C$ be a copermutad. Notice that (17) assemble into one map

[TABLE]

thus one has a degree $-1$ map $\sigma:\ \downarrow\mskip-3.0muC\to{\mathbb{P}}^{2}(\downarrow\mskip-3.0muC)$ defined as the composition

[TABLE]

with ${\mathbb{P}}^{2}(C)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,\uparrow^{2}\mskip-3.0mu{\mathbb{P}}^{2}(\downarrow\mskip-3.0muC)$ the obvious canonical isomorphism. Denote finally by ${\partial}_{\mathbb{\Omega}}$ the unique extension of $\sigma$ into a degree $-1$ derivation of ${\mathbb{P}}(\downarrow\mskip-3.0mu\hskip-0.50003ptC)$. One may verify by direct calculation that ${\partial}_{\mathbb{\Omega}}^{\,2}=0$, thus the following definition makes sense.

Definition 18**.**

The cobar construction of a copermutad $C$ is the differential graded (dg) permutad ${{\mathbb{\Omega}}}(C):=({\mathbb{P}}(\downarrow\mskip-3.0muC),{\partial}_{\mathbb{\Omega}})$. The dual bar construction of a permutad $A$ satisfying the assumptions of Example 16 is the dg permutad ${\mathbb{D}}(A):={\mathbb{\Omega}}(A^{*})$.

Convention. From now on we will assume that $A$ is a binary quadratic permutad as in Definition 12, with $B(1)$ finite dimensional.

Under the above assumption, one is allowed to define the Koszul dual copermutad by $A^{\hbox{\scriptsize\rm\raisebox{1.39998pt}{!`}}}:={A^{!}}^{*}$, the component-wise linear dual. With this notation, ${\mathbb{D}}(A^{!})={\mathbb{\Omega}}(A^{\hbox{\scriptsize\rm\raisebox{1.39998pt}{!`}}})$. As emphasized in [11], the copermutad $A^{\hbox{\scriptsize\rm\raisebox{1.39998pt}{!`}}}$ is more fundamental than $A^{!}$. It can be defined directly, without the passage through the permutad $A^{!}$, using coideals in cofree copermutads, without any assumptions on the size of the generators of $A$. Given the applications we had in mind, we however decided not to use this general definition and keep working with more intuitive ideals and free permutads.

Proposition 19**.**

The dual bar construction {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}) of \hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!} is isomorphic to the dg permutad $\hbox{\it CC}_{*}(P)$ of cellular chains of the permutohedron.

Proof.

The proof relies on the description of the permutad \hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!} given in Proposition 14. Denote by e_{n}\in\,\downarrow\mskip-3.0mu\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}(\underline{n})^{*}, $n\mskip-3.0mu\geq\mskip-3.0mu1$, the generator dual to \uparrow^{n-1}\hskip-5.0pt1\in\,\uparrow^{n-1}\mskip-3.0mu{\mathbb{k}}\cong\,\downarrow\mskip-3.0mu\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}(n). Then $e_{1},e_{2},\ldots$ are the free permutadic generators of {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}). Using the description of the structure operations of \hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!} given in Proposition 14, one easily verifies that the differential ${\partial}_{\,{\mathbb{D}}}$ in the dual bar construction {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}) is

[TABLE]

where $r_{1},r_{2}$ and the sign $\varepsilon(r)$ are as in (5), and \hbox{\tiny\diamondsuit}^{\mathbb{P}}_{r} is the structure operation of the free permutad {\mathbb{P}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}).

Referring to the description of $\hbox{\it CC}_{*}(P)$ given in Example 9, we define an isomorphism \xi:{\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\hbox{\it CC}_{*}(P) of free permutads by

[TABLE]

with $c_{n-1}$ the generator in (15). Comparing (18) to the formula for the differentials of $c_{n}$’s in [10, §9.3] we realize that they match up to an overall minus sign. Therefore $\xi$ defined above commutes with the differentials, thus constitutes the required isomorphism of dg permutads. ∎

Let us proceed towards the Koszulity property of binary quadratic permutads in Definition 12. One starts from a monomorphism $\uparrow\hskip-3.50006ptB\hookrightarrow A^{!}$ of collections defined as the composition

[TABLE]

Its linear dual is a surjection ${A^{!}}^{*}\twoheadrightarrow\ \uparrow\hskip-3.50006ptB$ which desuspens to a map $\pi:\ \downarrow\mskip-3.0mu{A^{!}}^{*}\twoheadrightarrow B$. The related twisting morphism555We borrowed this terminology from [11, Chapter 6]. $\downarrow\mskip-3.0mu{A^{!}}^{*}\to A$ is the composition

[TABLE]

It extends to a permutad morphism $\rho:{\mathbb{P}}(\downarrow\mskip-3.0mu{A^{!}}^{*})\to A$ by the freeness of ${\mathbb{P}}(\downarrow\mskip-3.0mu{A^{!}}^{*})$. One easily verifies:

Proposition 20**.**

One has $\rho\circ{\partial}_{\,{\mathbb{D}}}=0$, therefore $\rho$ induces the canonical map

[TABLE]

of dg permutads.

Definition 21**.**

A binary quadratic permutad $A$ is Koszul if the canonical map (19) is a component-wise homology isomorphism.

If $A$ is concentrated in degree [math], clearly $H^{0}({\mathbb{D}}(A^{!}))\cong A$, thus such an $A$ is Koszul if and only if ${\mathbb{D}}(A^{!})$ is acyclic in positive dimensions. This observation will be used in our proof of

Theorem 22**.**

The terminal permutad $\it per\hskip-1.00006pt{\mathcal{A}}s$ is Koszul.

Proof.

The statement follows from the isomorphism {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!})\cong\hbox{\it CC}_{*}(P) established in Proposition 19 combined with the contractibility of the permutohedron which implies that $\hbox{\it CC}_{*}(P)$ is acyclic in positive dimensions. ∎

Let us close this section by a Koszulity test in the spirit of the Ginzburg-Kapranov criterion for operads [8, Theorem 3.3.2]. For $A={\{A(\underline{n})\}_{n\geq 1}}\in{\tt Coll}$ with finite-dimensional pieces we define its generating power series as

[TABLE]

in which $\chi$ denotes the Euler characteristic. One then has

Proposition 23**.**

Assume that $A$ is a binary quadratic permutad as in Definition 12 with $B(1)$ finite dimensional. If $A$ is Koszul, then its generating series $f_{A}$ and the series $f_{A^{!}}$ of its Koszul dual are related by the functional equation

[TABLE]

Proof.

Suppose that $M={\{M(\underline{n})\}_{n\geq 1}}$ is a collection of graded vector spaces with finite-dimensional pieces. Simple combinatorics gives

[TABLE]

which is the same as

[TABLE]

Therefore the generating series of $M$ and that of the free permutad ${\mathbb{P}}(M)$ are related by

[TABLE]

When $A$ is Koszul, one has $f_{{\mathbb{D}}(A^{!})}=f_{A}$, while we already know from the above calculations that

[TABLE]

This finishes the proof. ∎

Example 24**.**

One has

[TABLE]

while

[TABLE]

Plugging it into (20) results in

[TABLE]

as expected.

Example 25**.**

A ‘twisted’ version of the terminal permutad $\it per\hskip-1.00006pt{\mathcal{A}}s$ of Example 13 is the quotient

[TABLE]

where $\mu\in{\mathbb{P}}(\mu)(\underline{1})$ is a degree [math] generator and $(\widetilde{S})$ the permutadic ideal generated by

[TABLE]

Notice that $\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s}$ equals the ‘parametrized associative permutad’ $q$-permAs of [10, §4.5] taken with $q=-1$. Since $\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s}$ is a permutadic version of the operad for antiassociative algebras which serves as a standard example of a non-Koszul operad [18, Section 5], one would expect that $\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s}$ is non-Koszul as well.

Surprisingly, it is not so. It turns out that that the dg collections {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s^{!}}}) and {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}) are isomorphic, though they are not isomorphic as dg permutads. This however suffices for the acyclicity of {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s^{!}}}) in positive dimensions, and thus the Koszulity of $\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s}$.

The isomorphism \zeta:{\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s^{!}}})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}) of dg collections is constructed as follows. As in the proof of Proposition 14 we establish that

[TABLE]

with the structure operations (10) the canonical isomorphisms

[TABLE]

without any additional sign factor. The calculation is actually even simpler than in the case of the untwisted $\it per\hskip-1.00006pt{\mathcal{A}}s$, since the relation induced by the ideal $(\widetilde{S}^{\perp})$ does not involve any signs. We infer that {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s^{!}}}) is freely generated by degree $n\mskip-3.0mu-\mskip-3.0mu1$ generators $\tilde{e}_{n}$, $n\geq 1$, whose differential is given by a formula analogous to (18) but without the $\varepsilon(r)$-factor. The underlying permutad of the dg permutad {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s^{!}}}) is then described as

[TABLE]

where $r_{i}$ is the cardinality of the set-theoretic preimage of $i\in\underline{k}$ via the map r:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k} . We have a similar formula for the underlying permutad of {\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}), namely

[TABLE]

The isomorphism \zeta:{\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\widetilde{\mathcal{A}}s^{!}}})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathbb{D}}(\hbox{\it per\hskip-1.00006pt{\mathcal{A}}s}^{!}) is, under the above identifications, given by

[TABLE]

with $\varepsilon(r)$ the sign of the shuffle associated to $r$. It is simple to verify that $\zeta$ defined this way commutes with the differentials of the dual bar constructions, so it is an isomorphism of dg collections. On the other hand, $\zeta$ is clearly not a morphism of permutads.

Part II Operadic category approach

4. Permutads as algebras over the terminal operad

The fundamental feature of Batanin-Markl’s (BM) theory of operadic categories [3] is that the objects under study are viewed as algebras over (generalized) operads in a specific operadic category, cf. also the introduction to [2]. Thus, for instance, ordinary operads are algebras over the terminal operad ${\sf 1}_{\tt RTr}$ in the operadic category ${\tt RTr}$ of rooted trees. The operad ${\sf 1}_{\tt RTr}$ is quadratic self-dual which, according to BM theory, implies that the bar constructions of its algebras (i.e. operads) are algebras of the same type (i.e. operads) again. As we noticed in the introduction, the same is true for permutads.

Let us start by recalling, following [2, §14.4], the operadic category ${\tt Per}$ that plays for permutads the same rôle as ${\tt RTr}$ for ordinary operads.666We refer to [3, §I.1] again for the language of operadic categories. Objects of ${\tt Per}$ are surjections \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}\in{\tt Surj}(n,k), $n\mskip-3.0mu\geq\mskip-3.0muk\geq\mskip-3.0mu1$, and morphisms $f:\alpha^{\prime}\to\alpha^{\prime\prime}$ of ${\tt Per}$ are diagrams

[TABLE]

in which $\gamma$ is order-preserving (and necessarily a surjection).

The cardinality functor is defined by |\alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}|:=k. The $i$-th fiber ${f}^{-1}(i)$ of the morphism (21) is the surjection (\gamma\alpha^{\prime})^{-1}(i){\hbox{ \twoheadrightarrow }}\gamma^{-1}(i), $i\in\underline{k}^{\prime\prime}$. The only local terminal objects of ${\tt Per}$ are the surjections

[TABLE]

which are also the chosen (trivial) ones. All quasi-bijections, and isomorphisms in general, are the identities.

Let us give also the following explicit description of the fiber ${f}^{-1}(i)$ of $f$ in (21). Assume that $k^{\prime}_{1}<\cdots<k^{\prime}_{t}$ are numbers such that

[TABLE]

and $n_{1}<\cdots<n_{s}$ are such that

[TABLE]

Then ${f}^{-1}(i)\in{\tt Per}$ is the surjection \underline{s}{\hbox{ \twoheadrightarrow }}\underline{t} that sends $u\in\underline{s}$ into the unique $j\in\underline{t}$ such that $\alpha^{\prime}(n_{u})=k^{\prime}_{j}$.

A ${\tt Per}$-operad ${\mathcal{P}}$ is a collection of vector spaces $\{{\mathcal{P}}(\alpha)\}_{\alpha\in{\tt Per}}$ equipped with the structure operations

[TABLE]

subject to the axioms listed in [3, Def. 1.11].

A particular example is the terminal ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$ with ${\sf 1}_{{\tt Per}}(\alpha):={\mathbb{k}}$ for each $\alpha\in{\tt Per}$, the operations (23a) the natural identification ${\mathbb{k}}\otimes{\mathbb{k}}^{\otimes k^{\prime\prime}}\cong{\mathbb{k}}$ and $\eta_{n}$ in (23b) the identity ${\mathbb{k}}={\mathbb{k}}$.777We are again slightly abusing the terminology; ${\sf 1}_{{\tt Per}}$ is in fact the linearization of the terminal ${\tt Per}$-operad in the Cartesian monoidal category of sets. Its algebras are described in

Proposition 26**.**

Algebras for the terminal ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$, in the sense of Definition 1.20 of [3], are permutads.

Proof.

The statement is a part of [2, Theorem 14.4] whose proof uses an explicit presentation of ${\sf 1}_{{\tt Per}}$, but we will show directly that ${\sf 1}_{{\tt Per}}$-algebras are the same as algebras for the monad ${\mathbb{P}}$ in Definition 7. Let $f:\alpha^{\prime}\to\alpha^{\prime\prime}$, $\alpha^{\prime}\in{\tt Surj}(n,k^{\prime})$, $\alpha^{\prime\prime}\in{\tt Surj}(n,k^{\prime\prime})$, be a morphism in ${\tt Per}$ with fibers $f_{1},\dots,f_{k^{\prime\prime}}$. The crucial fact on which our proof is based is that

[TABLE]

where $(-;{-,\ldots,-})$ is the substitution introduced in [10, Section 1.2]. Formula (24) follows directly from definitions, as the reader may check easily.

Let us inspect what [3, Definition 1.20] gives in our case. The underlying spaces of an algebra over an operad in an operadic category ${\tt O}$ is indexed by the set $\pi_{0}({\tt O})$ of its connected components, which is isomorphic to the set of chosen terminal objects. We identify

[TABLE]

thus ${\sf 1}_{{\tt Per}}$-algebras are collections $A=\{A(\underline{n})\in{\tt Vec}\ |\ \underline{n}\in{\tt Fin}\}\in{\tt Coll}$. By [3, Definition 1.20] again, the structure maps of an algebra $A$ over an operad ${\mathcal{P}}$ in an operadic category ${\tt O}$ are morphisms

[TABLE]

where $T\in{\tt O}$, $\pi_{0}(s(T))$ is the subset of $\pi_{0}({\tt O})$ formed by the connected components of ${\tt O}$ to which the fibers of the identity map $1\mskip-3.0mu\mskip-3.0mu1:T\to T$ belong, and $\pi_{0}(T)$ is the connected component of $T$. In our case, ${\mathcal{P}}$ is the constant ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$ whose each piece equals ${\mathbb{k}}$. If \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}\in{\tt Per}, clearly

[TABLE]

while $\pi_{0}(\alpha)=\underline{n}$. The structure maps associated to such an $\alpha$ are therefore given by

[TABLE]

According to [3, Definition 1.20], the structure maps (25) must assemble to a morphism $\mu:{\mathcal{P}}\to{\mathcal{E}}nd^{\,{\tt O}}_{A}$ of ${\mathcal{P}}$ to the endomorphism operad of the collection $A$. Let us inspect what it means in our case.

First of all, $\mu:{\sf 1}_{{\tt Per}}\to{\mathcal{E}}nd^{\,{\tt Per}}_{A}$ must preserve operad units. This means that for each $\alpha$ which is chosen local terminal, i.e. \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{1} for some $n\geq 1$, the diagram

[TABLE]

in which $\eta_{\alpha}$ is the unit morphism of the endomorphism operad, commutes. This is the same as to require that, for \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{1}, the structure map

[TABLE]

equals the identity. It thus bears no information, so we consider $m_{\alpha}$’s in (26) only for $|\alpha|\geq 2$.

Next, we must verify that $\mu:{\sf 1}_{{\tt Per}}\to{\mathcal{E}}nd^{\,{\tt Per}}_{A}$ commutes with the operadic structure operations. Assume therefore that $f:\alpha^{\prime}\to\alpha^{\prime\prime}$ is a morphism in (21), with fibers $f_{1},\dots,f_{k^{\prime\prime}}$. Its $i$th fiber $f_{i}$, $1\leq i\leq{k^{\prime\prime}}$, belongs to ${\tt Surj}(n_{i},l_{i})$, where $\underline{l}_{i}:=\gamma^{-1}(i)$ and $n_{i}\geq 1$ are such that

[TABLE]

Moreover,

[TABLE]

with some $n^{1}_{i},\ldots,n^{l_{i}}_{i}\geq 1$ such that

[TABLE]

while $\pi_{0}(f_{i})=\underline{n}_{i}$. Notice also that

[TABLE]

and that

[TABLE]

One easily finds that $\mu:{\sf 1}_{{\tt Per}}\to{\mathcal{E}}nd^{\,{\tt Per}}_{A}$ commutes with the structure operations of ${\tt Per}$-operads if and only if, for each $f$ as above, the diagram

[TABLE]

in which $\gamma_{f}$ is the operatic composition in ${\sf 1}_{{\tt Per}}$ and ${{\rm comp}}$ the obvious composition of linear maps, commutes. Since, in our case,

[TABLE]

and $\gamma_{f}$ is, under this identification, the identity, the commutativity of the above diagram is equivalent to the equation

[TABLE]

involving the structure maps (26). In other words, one requires that, whenever (24) is satisfied,

[TABLE]

which is an equality of maps

[TABLE]

To finish the proof, it is enough to realize how the substitution (24) enters, in the proof of [10, Proposition 1.4], the definition of the map $\Gamma_{E}$ in (14). The fact that ${\mathbb{P}}$-algebras are indeed the same as the structures described above will then be self-evident. ∎

5. Free ${\tt Per}$-operads

This section is a preparation for the construction of the minimal model of the terminal ${\tt Per}$-operad given in Section 6 and for the introduction of Koszul duality and Koszulity of ${\tt Per}$-operads given in Sections 7 and 8. Free operads were, in the context of general operadic categories, addressed in [2, Section 10] to which we refer for the terminology used in the following sentences. Our situation is however simplified by the fact that all local terminal objects in ${\tt Per}$ are the chosen terminal ones, thus unital ${\tt Per}$-operads are automatically strictly extended unital. Moreover, for our purposes it suffices to consider only $1$-connected ${\tt Per}$-operads, i.e. to operads ${\mathcal{P}}$ such that ${\mathcal{P}}(\alpha)\cong{\mathbb{k}}$ if $|\alpha|=1$.

With the above in mind, we introduce the category ${\tt Per}$-${\tt Coll}_{1}$ whose objects are collections $E=\{E(\alpha)\ |\ \alpha\in{\tt Per}\}$ of graded vector spaces with $E(\alpha)=0$ if $|\alpha|=1$, and level-wise morphisms. One has the obvious forgetful functor

[TABLE]

from the category of $1$-connected unital ${\tt Per}$-operads given by

[TABLE]

Theorem 10.11 of [2] guarantees that $\Box$ admits a left adjoint {\mathbb{F}}:\hbox{{\tt Per}$-${\tt Coll}{1}}\to\hbox{{\tt Per}$-${\tt Oper}{1}}. We call ${\mathbb{F}}(E)$ for E\in\hbox{{\tt Per}$-${\tt Coll}_{1}} the free ${\tt Per}$-operad generated by a $1$-connected ${\tt Per}$-collection $E$.

We describe in detail ${\mathbb{F}}(E)$ for generating collections of a particular form. The central rôles in this simplified construction will be played by the classical free non-$\Sigma$ operads. This generality is sufficient for all concrete applications given in the rest of this paper. The construction of ${\mathbb{F}}(E)$ for an arbitrary $1$-connected ${\tt Per}$-collection $E$ is sketched out at the end of this section.

Let ${\mathbb{\Delta}_{\rm semi}}$ be the subcategory of ${\tt Fin}$ with the same objects and order-preserving surjections as morphisms. It is an operadic category whose operads are the classical constant-free non-$\Sigma$ (non-symmetric) operads [3, Example 1.15]. One has a strict operadic functor, in the sense of [3, p. 1635], ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$ given on objects by {\rm des}(\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}):=\underline{k} while, for a morphism $f$ in (21), one puts ${\rm des}(f):=\gamma$. According to general theory [3, p. 1639], each ${\mathbb{\Delta}_{\rm semi}}$-operad ${\underline{{\mathcal{S}}}}$ determines,888We use the convention pioneered in [19] and distinguish classical non-$\Sigma$ operads by inderlyning. via the restriction along ${\rm des}$, a ${\tt Per}$-operad ${\rm des}^{*}({\underline{{\mathcal{S}}}})$ such that ${\rm des}^{*}({\underline{{\mathcal{S}}}})(\alpha)={\underline{{\mathcal{S}}}}({\rm des}(\alpha))$, $\alpha\in{\tt Per}$.

The form of the structure map $m_{f}$ of a ${\tt Per}$-operad ${\mathcal{P}}$ associated to a morphism $f:\alpha^{\prime}\to\alpha^{\prime\prime}$ in ${\tt Per}$ with fibers $f_{1},\dots,f_{k^{\prime\prime}}$ is recalled in (23a). For ${\mathcal{P}}={\rm des}^{*}({\underline{{\mathcal{S}}}})$, $m_{f}$ is determined by the structure operation $\underline{m}_{\gamma}:{\underline{{\mathcal{S}}}}(k^{\prime\prime})\otimes{\underline{{\mathcal{S}}}}(l_{1})\otimes\cdots\otimes{\underline{{\mathcal{S}}}}(l_{k^{\prime\prime}})\to{\underline{{\mathcal{S}}}}(k^{\prime})$ of ${\underline{{\mathcal{S}}}}$ associated to $\gamma={\rm des}(f)$999The form of structure operations of non-$\Sigma$ operads is recalled in (38) below. via the commutativity of the diagram

[TABLE]

Let $E=\{E(\alpha)\}_{\alpha\in{\tt Per}}$ be a collection of vector spaces such that

[TABLE]

and $\underline{E}={\{\underline{E}(\underline{k})\}_{k\geq 1}}\in{\tt Coll}$ be defined as

[TABLE]

for an arbitrary \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}, $k\geq 1$. Let finally $\underline{\mathbb{F}}(\underline{E})$ be the free non-$\Sigma$ operad generated by a collection $\underline{E}$ above, and ${\mathbb{F}}(E):={\rm des}^{*}(\underline{\mathbb{F}}(\underline{E}))$.

Proposition 27**.**

The ${\tt Per}$-operad ${\mathbb{F}}(E)={\rm des}^{*}(\underline{\mathbb{F}}(\underline{E}))$ is the free ${\tt Per}$-operad generated by the collection $E$. It is naturally graded,

[TABLE]

The grading is such that ${\mathbb{F}}^{1}(E)(\alpha)=E(\alpha)$ for all $\alpha\in{\tt Per}$, and

[TABLE]

Proof.

The claim is immediately obvious from the explicit description of free operads given in [2, Section 10], but we give an independent proof. Recall from [2, §5.2] that a strict operadic functor $p:{\tt O}\to{\tt P}$ is a discrete opfibration if

• (i)

$p$ induces a surjection $\pi_{0}({\tt O})\twoheadrightarrow\pi_{0}(\tt P)$ and

• (ii)

for any morphism $f:T\to S$ in $\tt P$ and any $t\in{\tt O}$ such that $p(t)=T$ there exists a unique $\sigma:t\to s$ in ${\tt O}$ such that $p(\sigma)=f$.

By dualizing [3, Theorem 2.4] one verifies that the restriction p^{*}:\hbox{\tt P-{\tt Oper}}\to\hbox{{\tt O}-{\tt Oper}} between the associated categories of operads has a right adjoint p_{*}:\hbox{{\tt O}-{\tt Oper}}\to\hbox{\tt P-{\tt Oper}} defined on objects by

[TABLE]

One thus has the adjunction

[TABLE]

It is easy to check that ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$ is a discrete opfibration, therefore the adjunction (30), with $\underline{\mathbb{F}}(\underline{E})$ in place of ${\mathcal{S}}$ and a non-$\Sigma$ constant-free operad ${\underline{{\mathcal{O}}}}$ in place of ${\mathcal{O}}$, gives

[TABLE]

Invoking the fact that $\underline{\mathbb{F}}(\underline{E})$ is the free non-$\Sigma$-operad, one sees that the right hand side of the above isomorphism consists of families of linear maps

[TABLE]

Since, by definition, ${\rm des}_{*}({\underline{{\mathcal{O}}}})(\underline{k})=\prod_{\alpha:\underline{n}\to\underline{k}}{\underline{{\mathcal{O}}}}(\alpha)$, taking into account the definition (28) of $E$, one sees that the family in (31) is the same as a family of linear maps

[TABLE]

In other words, ${\tt Per}$-operad maps ${\mathbb{F}}(E)\to{\underline{{\mathcal{O}}}}$ are in a natural one-to-one correspondence with families (32). This makes the freeness of ${\mathbb{F}}(E)$ obvious.

The free non-$\Sigma$ operad $\underline{\mathbb{F}}(\underline{E})$ is naturally graded, with $\underline{\mathbb{F}}^{1}(\underline{E})=\underline{E}$ [16, page 1475], and this grading clearly induces a grading of ${\mathbb{F}}(E)={\rm des}^{*}(\underline{\mathbb{F}}(\underline{E}))$ having the requisite properties. This finishes the proof. ∎

Let us point out that (27) characterizes collections E\in\hbox{{\tt Per}$-${\tt Coll}{1}} that are the restrictions of some collection \underline{E}\in\hbox{{\mathbb{\Delta}{\rm semi}}$-${\tt Coll}_{1}} along ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$, i.e. that are of the form $E={\rm des}^{*}(\underline{E})$ for some $\underline{E}$.

Remark 28**.**

As argued in [2, Subsect. 5.2], each discrete operadic opfibration $p:{\tt O}\to\tt P$ is obtained from a certain $\tt P$-cooperad ${\mathcal{C}}$ via the operadic Grothendieck construction. In our concrete case the corresponding ${\mathbb{\Delta}_{\rm semi}}$-cooperad ${\mathcal{C}}$ is given by

[TABLE]

General case. Let $E$ be an arbitrary $1$-connected ${\tt Per}$-collection. The constructions in Section 10 of [2] specialize to the description of the free ${\tt Per}$-operad ${\mathbb{F}}(E)$ generated by $E$ given below. We however need some notation.

For \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}\in{\tt Per} and $s\geq 1$, let ${\rm Tr}^{s}(\alpha)$ be the set of planar rooted trees growing from the bottom up with $s$ at least binary vertices, and leaves labeled from the left to right by the elements of the ordered set $\underline{k}$. We extend this definition by postulating ${\rm Tr}^{0}(\alpha):=\emptyset$ if $|\alpha|>1$, while for $|\alpha|=1$, ${\rm Tr}^{0}(\alpha)$ consists of a singular rooted tree with one leaf and no vertex.

Each vertex $v\in{\rm Vert}(\tau)$ of $\tau\in{\rm Tr}^{s}(\alpha)$ determines a segment $\underline{k}_{v}\subset\underline{k}$ of those $i\in\underline{k}$ for which $v$ lies on the path in $\tau$ connecting the leaf labeled by $i$ with the root. We then define $\underline{n}_{v}$ to be the ordinal given by the pullback

[TABLE]

Consider finally the surjection

[TABLE]

to the set ${\rm In}(v)$ of edges incoming to $v$ which sends $j\in\underline{n}_{v}$ to the unique edge in the path connecting $\alpha(j)$ to the root of $\tau$. The set ${\rm In}(v)$ is ordered by the clockwise orientation of the plane, and (33) is order-preserving. We will thus interpret it as an object of ${\tt Per}$, cf. the remark after diagram (1). We believe that Figure 1 makes these definitions clear.

One puts

[TABLE]

with

[TABLE]

For each morphism $f:\alpha^{\prime}\to\alpha^{\prime\prime}$ in ${\tt Per}$ with fibers $f_{1},\dots,f_{k^{\prime\prime}}$, a straightforward calculation reveals the existence of the canonical isomorphism

[TABLE]

which we take as the structure operation (23a) of the operad ${\mathbb{F}}(E)$. Notice that if $E={\rm des}^{*}(\underline{E})$ for some \underline{E}\in\hbox{{\mathbb{\Delta}{\rm semi}}$-${\tt Coll}{1}}, the above general construction coincides with the special one given in Proposition 27.

Remark 29**.**

If the reader finds this remark confusing, he or she may safely ignore it. The ${\tt Per}$-operad ${\mathbb{F}}(E)$ described above is the operad associated to the free Markl’s ${\tt Per}$-operad under the isomorphisms of the categories of ordinary and Markl’s operads stated in [2, Theorem 7.4] that holds due to the $1$-connectivity assumption; cf. also the notes at the beginning of Section 8.

Example 30**.**

It is easy to see that ${\mathbb{F}}^{0}(E)$ is as in (29) and ${\mathbb{F}}^{1}(E)\cong E$. To describe ${\mathbb{F}}^{2}(E)$ we call, following [2, Definition 2.9], a map $f:\alpha\to\beta\in{\tt Per}$ elementary if all its fibers except precisely one, say the $i$th fiber $F$, are trivial, i.e. the chosen terminal objects. We express this situation by writing $F\triangleright_{i}\alpha\stackrel{{\scriptstyle f}}{{\to}}\beta$ or simply $F\triangleright\alpha\stackrel{{\scriptstyle f}}{{\to}}\beta$ when $i$ is understood. We leave as an easy exercise to verify that, with this notation,

[TABLE]

where the summation runs over all elementary maps $f:\alpha\to\beta$ with $|\beta|\geq 2$.

6. Minimal model of the terminal ${\tt Per}$-operad and

homotopy permutads

Definition 31**.**

A minimal model of an operad {\mathcal{P}}\in\hbox{{\tt Per}$-${\tt Oper}_{1}} is a differential graded (dg) ${\tt Per}$-operad ${\mathfrak{M}}=({\mathfrak{M}},{\partial})$ together with a dg ${\tt Per}$-operad morphism $\rho:{\mathfrak{M}}\to{\mathcal{P}}$ such that

• (i)

the component $\rho(\alpha):({\mathfrak{M}}(\alpha),{\partial})\to({\mathcal{P}}(\alpha),{\partial}=0)$ of $\rho$ is a homology isomorphism for each $\alpha\in{\tt Per}$, and

• (ii)

the underlying non-dg ${\tt Per}$-operad of ${\mathfrak{M}}$ is free, and the differential ${\partial}$ has no constant and linear terms with respect to the natural grading of ${\mathfrak{M}}$ (the minimality condition).

Minimal models should be particular cofibrant replacements in a conjectural (semi)model structure on the category of ${\tt Per}$-operads. For the purposes of applications it however suffices to realize that minimal models are ‘special’ cofibrant [15, Definition 17]. This already guarantees that their algebras are homotopy invariant concepts.

Let \underline{a}:{\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}\,}_{\infty}\to\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}} be the minimal model of the terminal non-$\Sigma$ operad \underline{\hbox{\mathscr{A}\hskip-1.99997ptss}} governing associative algebras. As we know from [16, Example 4.8], {\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}\,}_{\infty} is generated by the collection $\underline{E}$ defined by

[TABLE]

with the differential acting on the generators by the formula

[TABLE]

where the summation runs over all $a,b\geq 0$ with $a\mskip-3.0mu+\mskip-3.0mub=r\mskip-3.0mu-\mskip-3.0mu1$, $1\leq i\leq a\mskip-3.0mu+\mskip-3.0mu2$, and where $\circ_{i}$ are the standard partial compositions in a classical operad. The dg operad morphism \underline{a}:{\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}\,}_{\infty}\to\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}} is given by

[TABLE]

where \mu\in\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}(2) is the generator of \underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}. Notice that {\sf 1}_{{\tt Per}}\cong{\rm des}^{*}(\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}). Define finally {\mathfrak{M}}:={\rm des}^{*}({\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}\,}_{\infty}), and $\rho:{\mathfrak{M}}\to{\sf 1}_{{\tt Per}}$ by $\rho:={\rm des}^{*}(\underline{a})$.

Theorem 32**.**

The dg ${\tt Per}$-operad map $\rho:{\mathfrak{M}}\to{\sf 1}_{{\tt Per}}$ defined above is a minimal model of the terminal operad ${\sf 1}_{{\tt Per}}$.

Proof.

The claim is almost obvious, but we still want to give some details, namely a formula for the differential ${\partial}$ in ${\mathfrak{M}}$. Given \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}\in{\tt Per}, $n\geq k\geq 1$, one has, by definition,

[TABLE]

while

[TABLE]

Under these identifications, the component $\rho(\alpha):{\mathfrak{M}}(\alpha)\to{\sf 1}_{{\tt Per}}(\alpha)$ of the map $\rho$ equals

[TABLE]

which is a homology isomorphism since {\underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}\,}_{\infty} is the minimal model of \underline{\hbox{\mathscr{A}\hskip-1.99997ptss}}. As a non-dg ${\tt Per}$-operad, ${\mathfrak{M}}$ is free, generated by the collection $E$ defined by

[TABLE]

where the $\xi$’s are the same as in (35).

Let us denote by $\xi_{\alpha}$ the replica of $\xi_{k-2}$ in $E(\alpha)$ above. Our next task will be to describe ${\partial}(\xi_{\alpha})\in{\mathfrak{M}}(\alpha)$. For natural numbers $k,a,b\in{\mathbb{N}}$ such that

[TABLE]

we define the map ${\gamma}^{a,b}_{i}:\underline{k}\to\underline{a\mskip-3.0mu+\mskip-3.0mu2}\in{\mathbb{\Delta}_{\rm semi}}$ by the formula

[TABLE]

Loosely speaking, ${\gamma}^{a,b}_{i}$ shrinks the interval $\{i,\ldots,i\mskip-3.0mu+\mskip-3.0mub\mskip-3.0mu+\mskip-3.0mu1\}\subset\underline{k}$ to $\{i\}$.

Let \alpha:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{k}\in{\tt Per}. By the opfibration property of ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$, there exists a unique \alpha^{a,b}_{i}:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{a\mskip-3.0mu+\mskip-3.0mu2}\in{\tt Per} and a unique morphism $f^{a,b}_{i}:\alpha\to\alpha^{a,b}_{i}$ such that ${\rm des}(f^{a,b}_{i})={\gamma}^{a,b}_{i}$. Explicit formulas for $\alpha^{a,b}_{i}$ and $f^{a,b}_{i}$ can be given easily.

All fibers of ${\gamma}^{a,b}_{i}$ are trivial (i.e. unique surjections to $\underline{1}$) except the $i$th one which we denote by $F^{a,b}_{i}$. Since ${\mathfrak{M}}(U)\cong{\mathbb{F}}(E)(U)\cong{\mathbb{k}}$ for trivial $U$’s as in (22), the structure map (23a) for ${\mathfrak{M}}$ corresponding to ${\gamma}^{a,b}_{i}$ reduces to the ‘partial composition’

[TABLE]

The formula for the differential then reads

[TABLE]

where the summation runs over $k,a,b\in{\mathbb{N}}$ as in (36) and $\xi_{\alpha^{a,b}_{i}}$ (resp. $\xi_{F^{a,b}_{i}}$) are the replicas of $\xi_{a}$ (resp. $\xi_{b}$) in $E(\alpha^{a,b}_{i})$ (resp. in $E(F^{a,b}_{i})$). Formula (49) makes the quadraticity of the differential ${\partial}$ in ${\mathfrak{M}}$, and therefore its minimality in particular, manifest. ∎

Formula (49) can be written in a more intelligent way. Recall from Example 30 that $F\triangleright_{i}\alpha\stackrel{{\scriptstyle f}}{{\to}}\beta$ expresses that $f$ is an elementary morphism whose only nontrivial fiber $F$ is the $i$th one. One then may rewrite (49) as

[TABLE]

where the summation runs over all elementary maps $F\triangleright_{i}\alpha\stackrel{{\scriptstyle f}}{{\to}}\beta$ such that $|\beta|\geq 2$. Following the philosophy of [16, Section 4], we formulate

Definition 33**.**

A strongly homotopy permutad is an algebra for the minimal model ${\mathfrak{M}}$ of ${\sf 1}_{{\tt Per}}$.

Remark 34**.**

Notice that to both (49) and (38) only partial compositions enter. This is because ${\mathfrak{M}}$ is isomorphic to the dual dg operad of the Koszul dual of ${\sf 1}_{{\tt Per}}$, cf.  the notes at the beginning of Section 8 and Corollary 49.

Strongly homotopy permutads can be described directly via their structure operations:

Proposition 35**.**

A strongly homotopy permutad is a collection $A={\{A(\underline{n})\}_{n\geq 1}}$ of dg vector spaces together with structure maps

[TABLE]

of degree $k\mskip-3.0mu-\mskip-3.0mu2$ defined for each $\alpha:\underline{n}\to\underline{k}$, $n\geq k\geq 2\in{\tt Per}$; here $\underline{n}_{i}:={\alpha}^{-1}(i)$, $1\leq i\leq k$. Moreover, for each such an $\alpha$, the equality

[TABLE]

is satisfied. Here the summation is the same as in (38), $\pi_{\beta}\circ_{i}\pi_{F}$ is the multilinear function obtained by inserting $\pi_{F}$ into the $i$th slot of $\pi_{\beta}$, and ${\partial}$ in the left hand side is the differential on the endomorphism complex induced by the differential of ${\{A(\underline{n})\}_{n\geq 1}}$.

Remark 36**.**

It can be easily checked that strongly homotopy permutads are algebras for the minimal model of the associative operad evaluated in the endomorphism operad that uses the shuffle product of vector spaces instead of the usual one. The similarity of ($P_{\alpha}$) with a formula in [16, Example 4.8] is therefore not surprising.

Example 37**.**

If $|\alpha|=2$, the sum in ($P_{\alpha}$) is empty, thus ${\partial}(\xi_{\alpha})=0$, so ${\partial}(\xi_{\alpha})$ is a dg map. If \alpha:\underline{m}{\hbox{ \twoheadrightarrow }}\underline{3}, the sum in ($P_{\alpha}$) has two terms, corresponding to the two possible order-preserving surjections \underline{3}{\hbox{ \twoheadrightarrow }}\underline{2}. The associated elementary morphisms are

[TABLE]

where $u,v,s,t\in{\tt Per}$ are as in Definition 5. Axiom ($P_{\alpha}$) now takes the form

[TABLE]

The degree [math] operations $\pi_{r}$ for $r\in{\tt Per}$ with $|r|=2$ are therefore of the same type as the operations \hbox{\tiny\diamondsuit}_{r} of Definition 5, but they satisfy the ‘associativity’ (11) only up to the homotopy $\pi_{\alpha}$. For $|\alpha|=4$, ($P_{\alpha}$) is a permutadic version of the Mac Lane’s pentagon.

Remark 38**.**

Let ${\underline{\mathcal{K}}}=\{{\underline{\mathcal{K}}}_{n}\}_{n\geq 1}$ be the cellular topological non-$\Sigma$ operad of the Stasheff’s associahedra [19, II.1.6]. Then ${\mathcal{M}}:={\rm des}^{*}({\underline{\mathcal{K}}})$ is the cellular topological ${\tt Per}$-operad such that the minimal model ${\mathfrak{M}}$ of ${\sf 1}_{{\tt Per}}$ is isomorphic to the associated cellular chain ${\tt Per}$-operad $\hbox{\it CC}_{*}({\mathcal{M}})$. This is where the ‘hidden associahedron’ of the title of this article hides.

Question 39**.**

The classical recognition theorem [20] states that a connected topological space has a weak homotopy type of a based loop space if and only if it is a topological ${\underline{\mathcal{K}}}$-algebra (aka $A_{\infty}$-space). Does there exist an analogous statement for topological ${\mathcal{M}}$-permutads?

As in the case of $A_{\infty}$-algebras, the left hand side of ($P_{\alpha}$) can be absorbed into the right one by interpreting ${\partial}$ as a structure operation. This can be done by allowing $\pi_{\alpha}$’s also for $|\alpha|=1$, i.e. for $\alpha$ the terminal object U_{n}:\underline{n}{\hbox{ \twoheadrightarrow }}\underline{1}, $n\geq 1$. The associated structure map then will be a degree $-1$ linear morphism

[TABLE]

Further, we need to allow in the sum of ($P_{\alpha}$) also trivial $F$ or $\beta$. The modified axiom reads

[TABLE]

For $\alpha=U_{n}$ it clearly gives $\pi_{n}^{2}=0$, thus $\pi_{n}$ is a degree $-1$ differential of $A(\underline{n})$. For a general $\alpha$, the left hand side of ($P_{\alpha})$ is absorbed in the right hand side of ($P^{\prime}_{\alpha}$) in terms with $|F|=1$ or $|\beta|=1$. We leave the details as an exercise.

7. Koszul duals of ${\tt Per}$-operads

A ‘classical’ operad is binary quadratic if it is generated by operations of arity $2$, and its ideal of relations is generated by relations of arity $3$. Each such an operad admits its Koszul, aka quadratic, dual, cf. [8, (2.1.9)] or [19, Definitions II.3.31 and II.3.37]. Similar notions exist for operads in a general operadic category [2, Section 11]. In the remaining two sections we will analyze the particular case of operads in ${\tt Per}$. The floor plan is similar to that of the parallel theory for permutads presented in Sections 2 and 3, so we can afford to be more telegraphic. We start with

Definition 40**.**

A ${\tt Per}$-operad ${\mathcal{P}}$ is binary quadratic if it is of the form ${\mathcal{P}}\cong{\mathbb{F}}(E)/(R)$, where

• (i)

the generators $E=\{E(\alpha)\}_{\alpha\in{\tt Per}}$ are such that $E(\alpha)=0$ if $|\alpha|\not=2$, and

• (ii)

the generators $R$ of the ideal of relations form a subcollection of ${\mathbb{F}}^{2}(E)$.

In Definition 40, ${\mathbb{F}}(E)$ is the free ${\tt Per}$-operad generated by the collection $E$. In concrete examples treated in the remainder of this section, $E$ will always be the restriction ${\rm des}^{*}(\underline{E})$ of some \underline{E}\in\hbox{{\mathbb{\Delta}{\rm semi}}$-${\tt Coll}{1}}, thus it may be, by the virtue of Proposition 27, realized by the restriction ${\rm des}^{*}(\underline{\mathbb{F}}(\underline{E}))$ of the free non-$\Sigma$ operad $\underline{\mathbb{F}}(\underline{E})$. The following notion is however recalled for an arbitrary $E$.

The Koszul dual ${\mathcal{P}}^{!}$ of a binary quadratic ${\tt Per}$-operad ${\mathcal{P}}$ [2, Definition 11.3] is the quotient

[TABLE]

where $\uparrow\hskip-3.50006pt\,E^{*}$ is the suspension of the component-wise linear dual of the generating ${\tt Per}$-collection $E$, and $R^{\perp}\subset{\mathbb{F}}^{2}(\uparrow\hskip-3.50006ptE^{*})$ is the component-wise annihilator of $R\subset{\mathbb{F}}^{2}(E)$ in the obvious degree $-2$ pairing between

[TABLE]

and

[TABLE]

here we use the explicit description of ${\mathbb{F}}^{2}(-)$ given in Example 30. The following statement follows from [2, Proposition 14.4] but we will present a proof that uses our explicit knowledge of the minimal model ${\mathfrak{M}}$ of ${\sf 1}_{{\tt Per}}$ acquired in Section 6.

Proposition 41**.**

The operad ${\sf 1}_{{\tt Per}}$ is binary quadratic. It is self-dual in the sense that the category of algebras over ${\sf 1}_{{\tt Per}}^{!}$ is isomorphic to the category of permutads via the functor induced by the suspension of the underlying collection.

Proof.

By Theorem 32, ${\sf 1}_{{\tt Per}}\cong H_{0}({\mathfrak{M}})$. Since ${\mathfrak{M}}$ is non-negatively homologically graded, its suboperad $Z_{0}({\mathfrak{M}})$ of degree [math] cycles equals the degree [math] piece ${\mathfrak{M}}_{0}$ of ${\mathfrak{M}}$. Clearly ${\mathfrak{M}}_{0}={\mathbb{F}}(E)_{0}\cong{\mathbb{F}}(E_{0})$ where, by the definition of the generating collection $E$,

[TABLE]

Likewise, ${\mathfrak{M}}_{1}={\mathbb{F}}(E)_{1}$ consists of compositions of some number of elements of $E_{0}$ and precisely one element of $E_{1}$, where

[TABLE]

Thus $B_{0}({\mathfrak{M}})={\rm Im}({\partial}:{\mathfrak{M}}_{1}\to{\mathfrak{M}}_{0})$ equals the ideal generated by ${\partial}(\xi_{\alpha})$, $|\alpha|=3$, where $\xi_{\alpha}$ is the replica of $\xi_{1}$ in $E(\alpha)$. Using formula (38) for the differential, we conclude that

[TABLE]

where $u,v,s$ and $t$ have the same meaning as in Example 37. This is the required binary quadratic presentation of the terminal ${\tt Per}$-operad. The natural pairing

[TABLE]

is given by

[TABLE]

In the above formula, the up-going arrow indicates the corresponding suspended generator. One thus immediately gets

[TABLE]

As in the proof of Proposition 26 we verify that ${\sf 1}_{{\tt Per}}^{!}$-algebras are collections $A={\{A(\underline{n})\}_{n\geq 1}}$ equipped with degree $+1$ operations

[TABLE]

satisfying the ‘anti-associativity’

[TABLE]

Let $\uparrow\hskip-3.50006ptA:={\{\uparrow\hskip-3.50006ptA(\underline{n})\}_{n\geq 1}}$ be the component-wise suspension of the collection $A$. It is straightforward to see that the operations

[TABLE]

make $\uparrow\hskip-3.50006ptA$ a permutad. The correspondence (A,\hbox{\tiny\diamondsuit}_{r}^{\uparrow})\mapsto(\uparrow A,\hbox{\tiny\diamondsuit}_{r}) is clearly an isomorphism between the categories of ${\sf 1}_{{\tt Per}}^{!}$-algebras and ${\sf 1}_{{\tt Per}}$-algebras. One may in fact show that the ‘operadic desuspension’ ${\mathbf{s}}^{-1}:{\sf 1}_{{\tt Per}}^{!}\to{\sf 1}_{{\tt Per}}$ defined by ${\mathbf{s}}^{-1}(\alpha):=\ \downarrow\mskip-3.0mu^{|\alpha|-2}$ is an isomorphism of ${\tt Per}$-operads. ∎

The following proposition will be formulated for any strict operadic functor [3, p. 1635] between arbitrary operadic categories, but the reader might as well consider only the case of ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$.

Proposition 42**.**

Let $p:{\tt O}\to{\tt P}$ be a strict operadic functor, ${\mathcal{P}}$ a ${\tt P}$-operad and ${\mathfrak{I}}\subset{\mathcal{P}}$ an ideal. Then

• (i)

the restriction $p^{*}({\mathfrak{I}})$ is an ideal in the ${\tt O}$-operad $p^{*}({\mathcal{P}})$.

• (ii)

If ${\mathfrak{I}}$ is generated by $G\subset{\mathfrak{I}}$, then $p^{*}({\mathfrak{I}})$ is generated by the restriction $p^{*}(G)$. Finally,

• (iii)

the restriction $p^{*}({\mathcal{P}}/{\mathfrak{I}})$ of the quotient ${\mathcal{P}}/{\mathfrak{I}}$ is isomorphic to $p^{*}({\mathcal{P}})/p^{*}({\mathfrak{I}})$.

Proof.

Items (i) and (ii) easily follow from the basic properties of operads in operadic categories and their ideals. Let us establish item (iii). Since the restriction $p^{*}$ is a functor \hbox{\tt P-{\tt Oper}}\to\hbox{{\tt O}-{\tt Oper}} between the categories of operads [3, p. 1639], applying it to the projection \pi:{\mathcal{P}}{\hbox{ \twoheadrightarrow }}{\mathcal{P}}/{\mathfrak{I}} leads to an operad morphism

[TABLE]

which is clearly surjective.

We want to prove that ${\rm Ker}(p^{*}(\pi))\cong p^{*}({\mathfrak{I}})$. Assume that $u\in p^{*}({\mathcal{P}})(t)$ for some $t\in{\tt O}$ is such that $p^{*}(\pi)(u)=0$. By the definition of the restriction, $p^{*}({\mathcal{P}})(t)={\mathcal{P}}(T)$ with $T:=p(t)$ and, likewise, $p^{*}({\mathcal{P}}/{\mathfrak{I}})(t)=({\mathcal{P}}/{\mathfrak{I}})(T)={\mathcal{P}}(T)/{\mathfrak{I}}(T)$. Under this identification, $p^{*}(\pi)$ acts as the projection {\mathcal{P}}(T){\hbox{ \twoheadrightarrow }}({\mathcal{P}}/{\mathfrak{I}})(T)=({\mathcal{P}}(T)/{\mathfrak{I}}(T)), so $u\in{\mathfrak{I}}(T))=p^{*}({\mathfrak{I}})(t)$. ∎

The following corollary can be stated for any discrete opfibration $p:{\tt O}\to{\tt P}$, but for our purposes the case of ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$ for which we recalled all the relevant notions will be sufficient.

Proposition 43**.**

Let $\underline{{\mathcal{P}}}$ be a binary quadratic non-$\Sigma$-operad. Then ${\rm des}^{*}(\underline{{\mathcal{P}}})$ is binary quadratic and, moreover, ${\rm des}^{*}(\underline{{\mathcal{P}}})^{!}\cong{\rm des}^{*}(\underline{{\mathcal{P}}}^{!})$

Proof.

Suppose that $\underline{{\mathcal{P}}}=\underline{\mathbb{F}}(\underline{E})/(\underline{R})$ is a binary quadratic non-$\Sigma$ operad. Then, by Proposition 27, ${\rm des}^{*}(\underline{\mathbb{F}}(\underline{E}))$ is the free ${\tt Per}$-operad ${\mathbb{F}}(E)$ on the restriction $E:={\rm des}^{*}(\underline{E})$ thus, by Proposition 42,

[TABLE]

with $R:={\rm des}^{*}(\underline{R})$. It is clear from definitions that ${\mathbb{F}}(E)/(R)$ is a binary quadratic presentation of the ${\tt Per}$-operad ${\mathcal{P}}:={\rm des}^{*}(\underline{{\mathcal{P}}})$. The second part of the proposition follows from the canonical isomorphisms

[TABLE]

which can be checked directly. ∎

8. Koszulity of ${\tt Per}$-operads

Operads are classically presented as collections ${\underline{{\mathcal{S}}}}=\{{\underline{{\mathcal{S}}}}(n)\}_{n\geq 1}$ of dg vector spaces with structure operations

[TABLE]

specified for any $k,n_{1},\dots,n_{k}\geq 1$. One also assumes a choice of the unit map $\eta:{\mathbb{k}}\to{\underline{{\mathcal{S}}}}(1)$. In [16, Def. 1.1] the author introduced an alternative definition, replacing the operations in (38) by the bilinear ones of the form

[TABLE]

In this setup, the grading of free operads is given by the number of structure operations applied to the generators minus one, so the operations in (39) naturally assemble into a single linear map

[TABLE]

This enables one to define (co)bar constructions of (co)operads by mimicking the analogous classical constructions for (co)algebras, cf. also the constructions for permutads in Section 3.

Partial compositions $\circ_{i}$ are expressed via the operations in (38) as

[TABLE]

whose vertical map is the product of the identities and the unit morphisms $\eta:{\mathbb{k}}\to{\underline{{\mathcal{S}}}}(1)$. For classical unital operads, both approaches are equivalent – the operations in (38) can be expressed via the $\circ_{i}$-operations as

[TABLE]

where $a\in{\underline{{\mathcal{S}}}}(k)$, $b_{i}\in{\underline{{\mathcal{S}}}}(n_{i})$ for $1\leq i\leq k$, and $\varepsilon=\pm 1$ is the Koszul sign of the permutation $(b_{1},\dots,b_{k})\mapsto(b_{k},\dots,b_{1})$. Operad-like structures based on ‘partial compositions’ as in (39) were later called Markl’s operads.

The arrays $n_{1},\dots,n_{k}$ indexing the operations in (38) are in one-to-one correspondence with order-preserving epimorphisms \gamma:\underline{m}{\hbox{ \twoheadrightarrow }}\underline{k}\in{\mathbb{\Delta}_{\rm semi}}; such a $\gamma$ determines the sequence $n_{1},\dots,n_{k}$ in which $n_{j}$ is the cardinality of the set-theoretic preimage $\gamma^{-1}(j)$, $1\leq j\leq k$. Notice that the definition of $\circ_{i}$ given in (41a) or (41b) uses $\gamma$’s of special types only, namely those for which the cardinality of the preimage $\gamma^{-1}(i)$ is $>1$ only for $j=i$. Such $\gamma$’s are prototypes of elementary maps in operadic categories.

Operads in general operadic categories also exist in two disguises which are, under favorable conditions, equivalent – in a form where the compositions in all inputs are made simultaneously; this is how they were introduced in [3] – and in Markl’s form where they are performed one after one. The crucial advantage of Markl’s form is, as in the classical case, that free Markl’s operads are naturally graded by the length of the chain of compositions and that the structure operations assemble into maps of the form (40a) for operads resp. (40b) for cooperads. This will be used in the definition of the central object of this section, the dual dg operad of a $1$-connected ${\tt Per}$-operad ${\mathcal{P}}$.

Recall from Example 30 that a map $f:\alpha\to\beta\in{\tt Per}$ in (21) is elementary if all its fibers except precisely one, say the $i$th fiber $F$, are trivial, i.e. equal to some local terminal objects $U_{n_{s}}$, $s\not=i$. This fact was recorded by $F\triangleright\alpha\stackrel{{\scriptstyle f}}{{\to}}\beta$. For each such an $f$ we define, mimicking (41b), the partial composition

[TABLE]

as the composite

[TABLE]

in which $m_{f}$ is the structure map (23a) and the vertical map is the product of the identities and isomorphisms ${\mathcal{P}}(U_{n_{s}})\cong{\mathbb{k}}$, $s\not=i$, which follow from the $1$-connectivity of ${\mathcal{P}}$. The partial compositions satisfy appropriate axioms [2, Definition 7.1] derived from the properties of $m_{f}$’s. The structure described above is Markl’s version of a ${\tt Per}$-operad ${\mathcal{P}}$. Under the $1$-connectivity assumptions, the categories of ${\tt Per}$-operads and Markl’s ${\tt Per}$-operads coincide [2, Theorem 7.4].

A Markl’s ${\tt Per}$-cooperad is a collection ${\mathcal{C}}=\{{\mathcal{C}}(\alpha\}_{\alpha\in{\tt Per}}$ with operations

[TABLE]

satisfying the dual versions of Markl’s operads [2, Definition 7.1]. As in Example 16, under obvious finitarity assumptions, the component-wise linear dual ${\mathcal{P}}^{*}$ of a Markl’s ${\tt Per}$-operad is a Markl’s ${\tt Per}$-cooperad. We will need also a reduced version of ${\mathcal{C}}$ whose underlying ${\tt Per}$-collection is defined by

[TABLE]

and its structure operation

[TABLE]

equals $\Delta_{f}$ in (43) if $|\beta|\geq 2$ while for $|\beta|=1$ it is the zero map

[TABLE]

In fancy language, ${\overline{\mathcal{C}}}$ is the coaugmentation coideal in Markl’s cooperad ${\mathcal{C}}$. It follows from (34) that the individual structure operations (44) assemble into a single map

[TABLE]

Definition 44**.**

A degree $s$ linear map $\varpi:{\mathcal{P}}\to{\mathcal{P}}$ of ${\tt Per}$-collections is a degree $s$ derivation of a Markl’s operad ${\mathcal{P}}$ if

[TABLE]

for every elementary $F\triangleright\alpha\stackrel{{\scriptstyle f}}{{\to}}\beta$ and the associated operation $\circ_{f}:{\mathcal{P}}(\beta)\otimes{\mathcal{P}}(F)\to{\mathcal{P}}(\alpha)$.

One easily sees that, given a $1$-connected ${\tt Per}$-collection, each degree $s$ linear map of ${\tt Per}$-collections $\zeta:E\to{\mathbb{F}}(E)$ uniquely extends to a degree $s$ derivation $\varpi$ of the free ${\tt Per}$-operad ${\mathbb{F}}(E)$. For a Markl’s ${\tt Per}$-cooperad ${\mathcal{C}}$ one has a degree $-1$ map $\zeta:\ \downarrow\mskip-3.0mu{\overline{\mathcal{C}}}\to{\mathbb{F}}^{2}(\downarrow\mskip-3.0mu{\overline{\mathcal{C}}})$ of ${\tt Per}$-collections defined as the composition

[TABLE]

where ${\overline{\Delta}}$ is as in (45) and ${\mathbb{F}}^{2}({\overline{\mathcal{C}}})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,\uparrow^{2}\mskip-3.0mu{\mathbb{F}}^{2}(\downarrow\mskip-3.0mu{\overline{\mathcal{C}}})$ the obvious canonical isomorphism

[TABLE]

Denote finally by ${\partial}_{\mathbb{\Omega}}$ the unique extension of $\zeta$ into a degree $-1$ derivation of ${\mathbb{F}}(\downarrow\mskip-3.0mu\hskip-0.50003pt{\overline{\mathcal{C}}})$. One easily verifies that ${\partial}_{\mathbb{\Omega}}^{\,2}=0$.

As we noticed at the beginning of this section, under the $1$-connectivity assumption, operads and Markl’s operads are just different presentations of the same objects, which is true also for (Markl’s) cooperads. We will therefore make no difference between them. Having this in mind, we formulate

Definition 45**.**

The cobar construction of a ${\tt Per}$-cooperad ${\mathcal{C}}$ is the dg ${\tt Per}$-operad ${{\mathbb{\Omega}}}({\mathcal{C}}):=({\mathbb{F}}(\downarrow\mskip-3.0mu{\overline{\mathcal{C}}}),{\partial}_{\mathbb{\Omega}})$. The dual dg operad of a ${\tt Per}$-operad ${\mathcal{P}}$ satisfying appropriate finitarity assumptions is the dg ${\tt Per}$-operad ${\mathbb{D}}({\mathcal{P}}):={\mathbb{\Omega}}({\mathcal{P}}^{*})$.

To introduce the Koszulity of a binary quadratic ${\tt Per}$-operad ${\mathcal{P}}$, one starts from an injection $\uparrow\hskip-3.50006ptE\hookrightarrow{\mathcal{P}}^{!}$ of ${\tt Per}$-collections defined as the composite

[TABLE]

Its linear dual ${{\mathcal{P}}^{!}}^{*}\twoheadrightarrow\ \uparrow\hskip-3.50006ptE$ desuspens to a map $\pi:\ \downarrow\mskip-3.0mu{{\mathcal{P}}^{!}}^{*}\twoheadrightarrow E$. As for permutads, one has the related twisting morphism $\downarrow\mskip-3.0mu{{\mathcal{P}}^{!}}^{*}\to{\mathcal{P}}$, which is the composition

[TABLE]

By the freeness of ${\mathbb{F}}(\downarrow\mskip-3.0mu{{\mathcal{P}}^{!}}^{*})$, it extends to a morphism $\rho:{\mathbb{F}}(\downarrow\mskip-3.0mu{{\mathcal{P}}^{!}}^{*})\to{\mathcal{P}}$ of dg ${\tt Per}$-operads. One verifies by direct calculation:

Proposition 46**.**

The morphism $\rho$ induces the canonical map

[TABLE]

of dg ${\tt Per}$-operads.

Definition 47**.**

A binary quadratic ${\tt Per}$-operad ${\mathcal{P}}$ is Koszul if the canonical map (47) is a component-wise homology isomorphism.

In the following theorem, ${\underline{{\mathbb{\Omega}}}}({\underline{{\mathcal{C}}}})$ denotes the cobar construction of a $1$-connected classical non-$\Sigma$ cooperad101010Notice that each $1$-connected cooperad is coaugmented. ${\underline{{\mathcal{C}}}}$, i.e. a non-$\Sigma$ version of the construction in [11, Section 6.5.2], and ${\underline{{\mathbb{D}}}}({\underline{{\mathcal{P}}}}):={\underline{{\mathbb{\Omega}}}}({\underline{{\mathcal{P}}}}^{*})$ the dual dg operad of a $1$-connected non-$\Sigma$ operad ${\underline{{\mathcal{P}}}}$ with appropriate finitarity properties that make the dualization possible.

Theorem 48**.**

Let ${\underline{{\mathcal{C}}}}$ and ${\underline{{\mathcal{P}}}}$ be as above, and ${\mathcal{C}}:={\rm des}^{*}({\underline{{\mathcal{C}}}})$, ${\mathcal{P}}:={\rm des}^{*}({\underline{{\mathcal{P}}}})$. Then

• (i)

the dg ${\tt Per}$-operads ${\mathbb{\Omega}}({\mathcal{C}})$ and ${\rm des}^{*}({\underline{{\mathbb{\Omega}}}}({\underline{{\mathcal{C}}}}))$ are isomorphic, as they are

• (ii)

the dg ${\tt Per}$-operads ${\mathbb{D}}({\mathcal{P}})$ and ${\rm des}^{*}({\underline{{\mathbb{D}}}}({\underline{{\mathcal{P}}}}))$.

• (iii)

Assume that ${\underline{{\mathcal{P}}}}$ is binary quadratic. Then the ${\tt Per}$-operad ${\mathcal{P}}$ is (binary quadratic) Koszul if and only if $\underline{{\mathcal{P}}}$ is one.

Proof.

The restriction along ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$ forms a functor from the category of dg ${\mathbb{\Delta}_{\rm semi}}$-collections to the category of dg ${\tt Per}$-collections. We already noticed that the restriction takes ${\mathbb{\Delta}_{\rm semi}}$-operads, i.e. non-$\Sigma$ operads, to ${\tt Per}$-operads. The same is true for Markl’s operads and cooperads. The restriction also commutes with (de)suspensions, component-wise linear duals and homology. By this we mean the following.

Recall that, if ${\underline{X}}$ is a rule that assigns to $\underline{n}\in{\mathbb{\Delta}_{\rm semi}}$ a dg vector space ${\underline{X}}(\underline{n})$, ${\rm des}^{*}({\underline{X}})$ assigns to $\alpha\in{\tt Per}$ a dg vector space ${\underline{X}}({\rm des}(\alpha))$. Then one has

[TABLE]

Let ${\underline{E}}$ be a ${\mathbb{\Delta}_{\rm semi}}$-collection, $\underline{\zeta}:{\underline{E}}\to\underline{\mathbb{F}}({\underline{E}})$ a linear map, and $\underline{\varpi}:\underline{\mathbb{F}}({\underline{E}})\to\underline{\mathbb{F}}({\underline{E}})$ the unique extension of $\underline{\zeta}$ into a derivation of the free non-$\Sigma$ operad $\underline{\mathbb{F}}({\underline{E}})$. Denote $E:={\rm des}^{*}({\underline{E}})$, $\zeta:={\rm des}^{*}(\underline{\zeta})$, and let $\varpi:{\mathbb{F}}(E)\to{\mathbb{F}}(E)$ be the unique extension of $\zeta$ into a derivation of the free ${\tt Per}$-operad ${\mathbb{F}}(E)$. Then

[TABLE]

The verification uses the obvious fact that ${\rm des}^{*}(-)$ brings derivations of non-$\Sigma$ operads into derivations of the corresponding ${\tt Per}$-operads, thus both ${\rm des}^{*}(\underline{\varpi})$ and $\varpi$ are derivations of ${\tt Per}$-operads extending the same linear map $E\to{\mathbb{F}}(E)$.

Let us prove (i). The underlying non-dg ${\tt Per}$-operads ${\rm des}^{*}(\underline{\mathbb{F}}(\downarrow\mskip-3.0mu{\overline{\underline{{\mathcal{C}}}}}))$ resp. ${\mathbb{F}}(\downarrow\mskip-3.0mu{\overline{\mathcal{C}}})$ of ${\rm des}^{*}({\underline{{\mathbb{\Omega}}}}({\underline{{\mathcal{C}}}}))$ resp. ${\mathbb{\Omega}}({\mathcal{C}})$ are isomorphic by Proposition 27. It remains to verify that ${\rm des}^{*}({\partial}_{{\underline{{\mathbb{\Omega}}}}})={\partial}_{\mathbb{\Omega}}$. The differential ${\partial}_{{\underline{{\mathbb{\Omega}}}}}$ in ${\underline{{\mathbb{\Omega}}}}({\underline{{\mathcal{C}}}})$ is, classically, the unique extension of the composition

[TABLE]

whose constituents have, as we believe, obvious meanings, into a degree $-1$ derivation of the free non-$\Sigma$ operad $\underline{\mathbb{F}}(\downarrow\mskip-3.0mu{\overline{\underline{{\mathcal{C}}}}})$. Likewise, we defined ${\partial}_{\mathbb{\Omega}}$ as the extension of the composition $\zeta:\,\downarrow\mskip-3.0mu{\overline{\mathcal{C}}}\to{\mathbb{F}}^{2}(\downarrow\mskip-3.0mu{\overline{\mathcal{C}}})$ in (46) into a derivation of ${\mathbb{F}}(\downarrow\mskip-3.0mu{\overline{\mathcal{C}}})$. It follows from (48a) and the obvious equality $\overline{\Delta}={\rm des}^{*}(\overline{\underline{\Delta}})$ that ${\rm des}^{*}(-)$ brings the chain of maps in (49) into the chain in (46), therefore ${\rm des}^{*}(\underline{\eta})=\eta$, thus ${\rm des}^{*}({\partial}_{{\underline{{\mathbb{\Omega}}}}})={\partial}_{\mathbb{\Omega}}$ by (48c).

To prove (ii), recall that ${\underline{{\mathbb{D}}}}({\underline{{\mathcal{P}}}})={\underline{{\mathbb{\Omega}}}}({\underline{{\mathcal{P}}}}^{*})$ by definition, therefore

[TABLE]

by (i), while

[TABLE]

by the first equality of (48b) and the definition of ${\mathbb{D}}({\mathcal{P}})$.

Let us finally attend to (iii). If ${\underline{{\mathcal{P}}}}$ is binary quadratic, then ${\mathcal{P}}$ is binary quadratic by Proposition 43. Recall [8, Definition 4.1.3] that the non-$\Sigma$ operad ${\underline{{\mathcal{P}}}}$ is Koszul if the canonical map111111The authors of [8] used slightly different conventions regarding the duals and suspensions, but our version, accommodated to the needs of the present article, is equivalent.

[TABLE]

induces an isomorphism of homology, while the Koszulity of ${\mathcal{P}}$ requires, by Definition 47, the same for the canonical morphism

[TABLE]

Under the identifications

[TABLE]

that follow from the second part of Proposition 43 and the already established item (ii) of Theorem 48, one clearly has that $\operatorname{{\it can}}={\rm des}^{*}({\underline{\operatorname{{\it can}}}})$ which, combined with the second equality of (48b), gives $H_{*}(\operatorname{{\it can}})={\rm des}^{*}(H_{*}({\underline{\operatorname{{\it can}}}}))$. Thus the map $\operatorname{{\it can}}$ is a homology isomorphism if

[TABLE]

is an isomorphism for each $\alpha\in{\tt Per}$. Since ${\rm des}:{\tt Per}\to{\mathbb{\Delta}_{\rm semi}}$ is an epimorphism on objects, this happens if and only if $H_{*}({\underline{\operatorname{{\it can}}}}(\underline{n}))$ is an isomorphism for each $\underline{n}\in{\mathbb{\Delta}_{\rm semi}}$, i.e. when ${\underline{\operatorname{{\it can}}}}$ induces an isomorphism of homology. ∎

Corollary 49**.**

The terminal ${\tt Per}$-operad ${\sf 1}_{{\tt Per}}$ is Koszul.

Proof.

One way of proving the statement would be to identify ${\mathbb{D}}({\sf 1}_{{\tt Per}}^{!})$ with the minimal model ${\mathfrak{M}}$ described in Proposition 32. The corollary however follows from Theorem 48 since ${\sf 1}_{{\tt Per}}$ is the restriction of the terminal non-$\Sigma$ operad $\underline{{\mathcal{A}}{\it ss}}$ whose Koszulity is superclassical. ∎