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In the paper βEzra Getzler, M. M. Kapranov, Cyclic operads and cyclic homologyβ (linked at cyclic operad), I find the following definition of a monad π on symmetric sequences whose algebras are supposed to be operads:
π(V)=β¨TV(T)where V(T)=β¨vβv(T)V(in(v)), with v(T) the vertices of the tree T and in(v) the set of incoming edges at v.
I am confused about the β¨ in the definition of π(V). In their equation (1.9) it is labeled as over βrooted n-trees Tβ but in the sentence preceeding it is said to be over βisomorphism classes of rooted treesβ. But I canβt make sense of either one. If it is over all trees, then there is a lot of duplication; but if it is over isomorphism classes of trees, then what does it mean to talk about βthe set of verticesβ of an isomorphism class of trees?
Honestly, Iβve found that article pretty confusing in the past.
Now I am looking at our page free operad which has a similar formula, summing over isomorphism classes of trees:
β[T]βΟ0(Trees)ΛK(T)βAut(T)ΛΞ»(T)As written, this has the same problem that things like ΛK(T) and Aut(T) are not defined if we are given only an isomorphism class of trees [T]. Perhaps what is meant is to choose a representative of each isomorphism class? But even in this case, the two formulas donβt agree: the second one is βtensored upβ to get a Ξ£n-action.
Choosing representatives also makes me uneasy because we have to define the multiplication of the monad by βgrafting treesβ, and that generally wonβt produce the chosen representative; so it seems we would have to choose an isomorphism from the result to our chosen representative of its equivalence class, and this wonβt be coherent. But perhaps the earlier definition as a coend β«TβTreesΛK(T)βΛΞ»(T) avoids this problem, and the point is that the version involving isomorphism classes is only related to the βrealβ definition by a noncanonical isomorphism so we canβt transfer the monad structure to it?
I have several half-written articles on my public nLab web and my private nLab web which goes into these matters. Somewhat in the manner of Richard Williamsonβs series of posts which walk through a development, Iβll try the same thing here, drawing on one of the articles in my private web.
We let FB denote the symmetric monoidal groupoid of finite sets and bijections, where the monoidal product is obtained by restricting the coproduct on the category of finite sets. This is equivalent as a symmetric monoidal category to the permutation category P, the free symmetric monoidal category on one generator.
Let F,G:FBβV be two species valued in a complete cocomplete symmetric monoidal closed category V. The graft product FβG is defined by the formula
(FβG)[S]=βTβSF[S/T]βG[T]where S/T denotes the pushout of Tβ1 along the inclusion TβS. This includes the degenerate case where T=0; in this case S/0 is the result of freely adjoining a point to S.
The set S/T has T/T as distinguished basepoint, and if U is complementary to T, we have a natural bijection S/Tβ U+{T/T}. It follows that
(FβG)[S]β βU+T=SF[U+*]βG[T]β βU+T=SFβ²[U]βG[T]β (Fβ²βG)[S]where β refers to the usual Day convolution product, induced from the tensor product on FB, and Fβ² is the derivative of the species F, defined by the formula Fβ²[S]=F[S+*]. Since differentiation of species
V(β)+*:VFBβVFBis cocontinuous, and since Day convolution is separately cocontinuous (i.e., cocontinuous in each of its arguments), we see that
FβGβ Fβ²βGis also separately cocontinuous.
For V=Set, a structure of species FβG is given by three data:
In this post, Iβll make some remarks on the βlax monoidalβ structure of the graft product.
The graft product is not associative up to isomorphism, but we do have a βlaxβ associativity. Specifically, we calculate
(FβG)βHβ (Fβ²βG)β²βHβ Fβ³βGβH+Fβ²βGβ²βHβ Fβ³βGβH+Fβ(GβH)so that there is a noninvertible associativity
Ξ±FGH:Fβ(GβH)β(FβG)βHgiven by inclusion, natural in each of its arguments F, G, and H, and satisfying an evident pentagon coherence condition. Additionally, there is a lax unit, defined as the species X for which X[S] is the monoidal unit of V if S is a 1-element set, else X[S] is initial. Here we have evident natural maps
Ξ»F:XβFβF,ΟF:FβXβFThe first map Ξ»F is invertible, but the second is not: its component at a finite set S is the codiagonal
β:Sβ F[S]βF[S]However, all coherence conditions for monoidal categories hold (I wonβt prove this here). We may call such a structure, relaxing the condition of invertibility of Ξ± and Ο but retaining the naturality and coherence conditions, a lax monoidal category.
In the sequel, Fβn will denote the iterated graft product defined recursively by
Fβ0=X,Fβ(n+1)=FβnβFso that all parentheses in Fβn are to the left. We have a partial coherence theorem:
Proposition: Any two maps
FβmβFβnβFβ(m+n)definable in the language of lax monoidal categories are equal. We denote this map by Ξ±mn.
For F a Set-valued species, an element of Fβn is described by data as follows:
A structure of tree constructed from n sprouts in a prescribed plugging order;
An element of F[Ο] for each sprout Ο used in the construction of the tree in 1.
Thus we expect some connection between the βgeometric seriesβ βnFβn and the free operad on F. The only difference between the two is that the standard construction of free operads via trees is in no way dependent on plugging order. Thus, to obtain the free operad, we have to mod out by isomorphisms on Fβn induced by βinessentialβ differences in plugging order. This will be described in detail in the section on trees and hierarchies.
In this comment will be a brief interlude on presenting the notion of operad.
There are various ways to describe operads and free operads. For example,
An operad C is a monoid in the category of species with respect to a monoidal product known as the substitution product (denoted β). When unpacked, this means we have a collection of maps
C[k]βC[n1]ββ¦βC[nk]βC[n1+β¦+nk]and an identity element IβC[1], satisfying appropriate associativity and unit conditions.
Alternatively, an operad consists of objects C(k) and a collection of morphisms
plugi:C[j]βC[k]βC[j+kβ1]satisfying suitable properties. In terms of sets, where elements pβC(j) are interpreted as j-ary operations XjβX, the meaning of plugi(p,q) is that outputs of the operation q are plugged in as inputs for the ith argument of p, resulting in an operation p(1,β¦,q,β¦,1).
Each description has its advantages. The first is concise and fits into clean categorical algebra. The second is easier to use for describing things like bar or cobar resolutions.
The description in terms of graft products, given below, is essentially a way to repackage the second description; the plugging operations
plug1,β¦,plugare rolled together into a single operation
one for each pair , and then the operations are then rolled together into a single species map . This map is to satisfy various associativity and unit conditions. With some qualification (related to mere lax associativity of the graft product)
A key point for us is the following analogy. In one of the standard bar resolutions for groups or monoids , one starts with the augmentation ideal defined by an exact sequence
and then one defines a chain complex with components , or in other words a differential on where the tensor algebra
is seen as a graded algebra. Part of what makes this mechanism succeed, in particular the description of the tensor algebra, is the fact that the tensor product is separately continuous in its arguments.
One could try to develop a bar resolution for operads by analogy, and in fact that is what one does, but not in a simple-minded way suggested by the first description of operads. That is, one does work with a resolution based on a free operad (free operads being analogous to tensor algebras) generated by a species analogue of an augmentation ideal, but the graded components are not iterated substitution products , and indeed, the free operad on a species is not formed in a simple-minded way as
because that is false (it would be true if the monoidal product were separately cocontinuous, or even if it preserves countable coproducts in its separate arguments).
Instead, in the literature, the bar resolution for operads is developed more in accordance with the second description of operads. It is not exactly based on a geometric series
(because that is still not quite right for the free operad), but itβs based on something close to that, as will be explained below. In any event, it is relevant for this development that is separately cocontinuous in each of its arguments.
Note that in this way of developing operad theory, the notion of derivative of a species plays a more visible role. This is perhaps fortunate not just because we understand the calculus of species as a kind of categorification of the differential calculus, but also because it brings to light phenomena that usually go unremarked. Some examples:
The derivative of an operad is naturally an algebra, in such a way that the operad structure according to the second description,
endows with a natural structure of -module.
The derivative of a cyclic operad is an ordinary operad.
Now weβll go a little deeper into the structure of free operads, by analyzing further the structure of in terms of (for us, useful) notions of hierarchies and trees.
It is interesting to resolve into a sum of monomials, each of which is a tensor product of derivatives of . For example, we have
There are in fact monomial terms in , each a tensor product of higher derivative terms, and each corresponding to a function (here ) such that for all . Think of such a function as specifying a βplugging orderβ, where each tensor factor occurring in a monomial is plugged into a vertex marked by a derivative symbol which occurs prior in the monomial term.
Definition: Let be the set of integers between and . A hierarchy (of order ) is a function such that for all . If is a pointed set, then a rooted tree on is a morphism of pointed sets such that every is taken to the basepoint by some iterate of : for all there exists such that .
Note there are possible hierarchies on the -element set . We denote this set by , or rather .
There is an evident notion of morphism and isomorphism of rooted trees. Evidently every hierarchy gives a rooted tree structure on the set with as basepoint. There is thus a groupoid consisting of hierarchies on and isomorphisms between them. The disjoint union of groupoids forms a full subgroupoid of the groupoid of finite rooted trees. In fact, the inclusion
is essentially surjective, hence an equivalence.
For a set , define the derivative of a species by the formula . For each hierarchy of order , define a species by
Proposition: If is a species, then
for every .
Proof: We proceed by induction. The case is clear. We have
by the product rule for differentiation. Then, to each summand indexed by , we associate a new hierarchy defined by for , and . This means we can rewrite the last sum as
and we are done.
Next, let be the set of permutations on . If defines an isomorphism between hierarchies of order , then there is an isomorphism (natural in species )
and accordingly each species gives rise to a representation of the groupoid .
Example: Consider the hierarchy that takes both and to . There is an automorphism from to itself, given by the simple transposition interchanging and . The corresponding automorphism on , evaluated at an object , is a coproduct of evident isomorphisms
ranging over the possible decompositions . Thus the induced automorphism involves transport of -structures along bijections of sets (as in the map
where swaps and ), in addition to corresponding permutations of tensor factors, where we permute the factors
Now for a final comment before taking a breather. Here we describe the free operad.
Write for the functor that takes a species to the representation of the groupoid . Thus defines a functor
An alternative view is that there is a fibering
which maps each monomial term in down to a copy of the terminal species , indexed by the appropriate hierarchy in . This fibering may be regarded as a span from to in . The groupoid may be regarded as a monad on in that acts on via the representation .
Eventually we will prove the following theorem.
Theorem: Let denote the colimit functor
Then sends a species to the free operad on . In alternative notation,
This will be explored further in comments to come, perhaps.
Wow! What an impressive picture.
I have one silly question: way back up in #4, isnβt the Day convolution a coend rather than a coproduct? Is that coend getting wrapped up somewhere else in this presentation?
The reason Iβm asking about this is that I wanted to understand the analogous construction for cyclic operads, and then generalize it to cyclic multicategories. You mentioned cyclic operads at one point, so I suppose youβve thought about how it works for them too; Iβd be interested to hear whatever you have to say about them.
And have you thought about how it works for multicategories (colored operads)? Presumably gets replaced by the free symmetric strict monoidal category on the set of objects (colors)?
Hi Mike; thanks for your feedback!
That the tensor product , which abstractly is indeed a Day convolution coend induced from the monoidal structure of , can be reformulated as a sum
is something I first learned from the Joyal theory of species papers. The idea is that from , each bijection induces a decomposition of into two components , , so that the coend may be rewritten
where the last isomorphism comes about by the Yoneda (or coYoneda) lemma.
(If you havenβt gone through those species papers of Joyal and his coworkers (like Labelle), they are highly recommended! Really beautiful stuff.)
Iβll lead up to some ruminations on cyclic operads in the next couple or three comments to come (Williamson-like), but to be honest I havenβt pursued those thoughts as hard as I ought to have. Mostly these are things which came out of considering the Lie operad, which is truly a key example.
(No, I havenβt thought much about the colored case.)
Thanks! Now that you describe it, it kind of rings a bell, but it still seems weird to me, like something that shouldnβt work. (-: Does that work for Day convolution over any, um, cocartesian monoidal extensive category?
Without thinking about it very hard, it might work for the groupoidal core of a cocartesian monoidal extensive category.
Edit: And maybe for a cocartesian monoidal extensive category as well, although Iβve had a few drinks. If true, that would seem potentially interesting for a more Lawvere theory context, as opposed to an operad context.
Probably what I meant was the groupoidal core.
Let me pick up again from the end of #8. I now want to reformulate the notion of permutative -operad in terms of the graft product .
Recall that for a species , we have the decomposition
where is the hierarchy sending to , and is the hierarchy considered at the end of #7, sending both and to . The inclusion of in is identified with the lax associativity .
Definition: An -valued operad is a species equipped with maps , (recall from #5; it is Joyalβs notation) such that the following conditions hold.
The restriction of to is given by
In other words, the diagram
commutes;
The restriction of to coequalizes the pair of automorphisms induced by the two hierarchy automorphisms on , namely the identity automorphism and the automorphism on described in the Example at the end of #7.
The composite
is , and the composite
is . (See #5 for the descriptions of the left and right unit constraints and .)
The first two conditions correspond to the associativity condition in the usual definition of operad (monoid with respect to plehystic monoidal product on species β what Joyal called the substitution product and denotes by ), and the last condition to the unit conditions.
This reformulation is much closer to the idea of operad that is implicit in Stasheffβs Homotopy of H-Spaces papers, where for example the associahedra carry operations for , which we called in #6 above and which we have rolled into a single species map . Also this reformulation is the right one to use if we want to consider non-unital operads as Stasheff did (he considered associahedra but no ); if we define a non-unital operad instead as a species equipped with an associative multiplication , we get something much too weak or inexpressive for this purpose.
Before we prove that these conditions are necessary and sufficient, Iβd like to observe that if is an operad (Γ la the definition above), then its derivative naturally carries a structure of algebra (i.e., monoid with respect to the convolution product ), something I alluded to at the end of #6. For if we take the derivative of the species map with the help of the product rule, we get a map
which in turn given by a pair of maps , . By taking the derivative on the first associativity condition, one sees that the first of these maps is an associative operation. The derivative of the operad unit gives a unit where is the monoidal unit for , and the third condition implies that satisfies the unit axioms for an algebra. Finally, note that the map endows with a structure of module over this algebra ; again the requisite axioms for a module follow from the first and third conditions of the definition.
Iβll give some examples of this later, perhaps. An important example is the Lie algebra operad as a -enriched operad, whose derivative is the tensor algebra species whose algebra structure as free noncommutative algebra coincides with the derivative structure described above.
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