Blog Post number 1

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N-operads and associahedra

June 16, 2023

Defining an N-operad using transfer systems

This time, we want a combinatorial understanding of N-operads in a suitable G-equivariant setting. The goal is establishing, for G=Cpn, a one-one correspondence between the natural order on the collection of N-operads and the poset structure of Stasheff’s (n+1)-associahedron, An+1. This is work by Balchin-Barnes-Roitzheim1

.

Instead of the traditional way, as mentioned right at the article’s outset, we want to understand N-operads differently in a combinatorial way. The key player is the following result that was conjectured by Blumberg-Hill 2

and proved by a series of works by Bonventre-Pereira, Gutiérrez-White, and Rubin independently. I just state the result by getting into the details of the proof

Thm 1.1. 

\[\text{Ho}\leftMissing or unrecognized delimiter for \right \xrightarrow{\sim} \mathfrak{Ind}\]

where N-op is the category of N-operads, and Ind is the category of indexing systems with poset structure.

Now we’ll see what indexing systems are next.

For that, they’re some machinery that we introduce first.

Def 1.2.  A categorical coefficient system is a contravariant functor

C_:OrbGCat

from the orbit category of G to the category of small categories.

Typically, there’s an abuse of notation by omitting the word categorical.

We have the notion of a symmetric monoidal coefficient system, which is a contravariant functor from the orbit category of G to the category of symmetric monoidal categories and strong symmetric monoidal functors.

The category of interest for us is the coefficient system of finite G-sets.

Def 1.3.  Suppose Set is the symmetric monoidal coefficient system of finite sets, with Set(H)=SetH. The symmetric monoidal operation is the disjoint union.

Now the idea is to associate to every N-operad a sub coefficient system of Set.

Next is the definition of an indexing system using all the machinery we’ve introduced.

Def 1.4.  An indexing system is a sub symmetric monoidal coefficient system that’s closed under subobjects and self-induction.

Now the latter condition, for a full sub-symmetric monoidal coefficient system Z, translates to the following.

\[H/H' \in \mathcal{Z}H \wedge K \in \mathcal{Z}H \Rightarrow H \times_{H'} K \in \mathcal{Z}H\]

We now compare indexing systems to transfer systems

.

Lemma 1.5.  An indexing system determines andisalsodeterminedby a set ZH for HG, with subgroups of H satisfying the following axioms:

  • H/HZ(H)

  • H/HZ(H)gHg1/gHg1gZg1

  • H/HZ(H)  H/KZ(h)H/KZ(H)

  • H/HZ(H)K(KH)K , KH

All this data is referred to as a transfer system.

Alternatively, we have the following definition.

Def 1.6.  For a transfer system Z(H) we can define a set abstractly

\[\{ N_{H'}^{H} \mid H/H' \in \mathcal{Z}H, H \leq G \}\]

We call these norm maps.

Using all these, now we have an alternate description of an N-operad that I state next.

Cor 1.7. 

G is a finite group. An N operad for G, upto homotopy is defined as the data of the set X={NKH1K<HG} satisfying the following andallconjugatesthereof

  • NKHX M<HNMKMX (Restriction Axiom aka RA)
  • NKLX NKHXNHLX (Composition Axiom aka CA)

It’s important to note that the notation choice makes sense so that the Mackey functor of an algebra A over an N-operad O has multiplicative norm maps NHK:πK0(A)πH0(A).

The case of G=Cpn

We now look at the case of interest, i.e. when G=Cpn where p is a prime.

The goal is to understand the sequence {N(Cpn)} .

The cases of n=0 and n=1 are trivial. We look at the case of n=2.

We start by drawing out all our possibilities and then using the axioms RA and CA to eliminate redundant structures.

The third structure i.e. {N10,N21} isn’t an N-structure using CA and the sixth and eighth structures i.e. {N20,N21} , {N20} aren’t N-structures by an application of RA.

Thus we can conclude that N(Cp2)∣=5

There’s a noticeable pattern that we’ve here.

We now spend some time recalling the Catalan Numbers, which obviously have been popping up so far.

The Catalan sequence is a widespread sequence popping up in various enumerations problems, two of which we’ll use directly. We first define the sequence.

Def 2.1.  The Catalan numbers are terms of the sequence {Cn} defined as follows:

\[C_n = \frac{1}{n+1}\binom{2n}{n} = \frac{2n!}{n+1!n!}\]

We also recall that they satisfy the following recurrence relation.

C0=1
\[C_{n+1} = \sum_{l=0}^{n}C_l C_{n-l} \ , \ n\geq 0\]

Some examples of enumeration problems where Catalan numbers pop up are the following:

  • Cn= # rooted binary trees with n+1 vertices
  • Cn= # ways of cutting a convex polygon with n+2 sides into triangles
  • Cn= # permutations of {1,,n} with no three terms in increasing order

BBR product

We now define an operation on N-operads that I call the Balchin-Barnes-Roitzheim BBR product, also called the o-dot product.

\[\odot: N_{\infty}Cpk \times N_{\infty}Cpl \rightarrow N_{\infty}Cpk+l+2\]

For XN(Cpk) and XN(Cpl), XXN(Cpk+l+2) is defined as follows:

\[X \odot X' = X \coprod \Sigma^{k+2}X' \coprod \{N_{k+1}^{m} \}_{k+1 < m \leq k+k+2}\]

where ΣjNba=Nb+ja+j

Now there’s a pictorial way to work and see things that we see next.

BBR product
Pictorial illustration of the BBR product from the BBR paper

We now see an example:

Let X,YN(Cp2)

BBR product example
It’s easy to see how the operation works and that it’s not commutative

We now see the result that lets us use the BBR product for our purposes.

Prop 3.1.  XN(Cpk) and YN(Cpl). XY is an object in N(Cpk+l+2) for k,l1. Also conversely, if we’ve XYN(Cpk+l+2) then X and Y are N structures.

Now we make our next step towards computing N(Cpn)

Prop 3.2.  We have the following recurrence relation

\[|N_{\infty}Cp1 | = 1\] \[|N_{\infty}Cpn| = \sum_{q=0}^{n}|N_{\infty}Cpq1| |N_{\infty}Cpnq1|, \ n \geq 0\]

The idea of the proof is to use the last proposition after showing that every XN(Cpn) can be written as X=YZ where YN(Cpq1) and ZN(Cpnq1)

Now we begin with jZ such that it’s the minimum integer such that Njn is in X. The cases of j=0 and j=n are straightforward, with Y= and Z= in both cases.

Now if O<j<n, then clearly we can split off Cpi with 0i<j. Let’s call this part YN(Cpj1) and the remaining part X Now it’s easy to see that this part’s like Z for some ZN(Cpnj1).

We can thus conclude that X=YZ

Now a consequence of the previous result is the following corollary.

Cor 3.3. 

For G=Cpn, and XN(Cpn) we can write X=YZ for some suitable N-operads Y and Z.

Now as a corollary to this, we have the first part of the main theorem.

Cor 3.4.  We have a bijection

\[\{\text{rooted binary trees with} \ n+2 \ \text{vertices} \} \Leftrightarrow \{N_{C_{p^n}} \}\]

The recursive construction works as follows:

Same cases Then

Same cases Relation to Stasheff's Associahedra ------ We now define the associahedron. For binary trees, let's define the following relation: For binary trees X and Y X<YY can be obtained fromXby means of a finite sequence of rotations

Def4.1. The poset structure on binary trees with n+1 vertices induced by the above relation is referred to as Stasheff's n-associahedron An. We now equip N-operads for G=Cpn with a poset structure by defining the following order relation: For X,YN(Cpn) X<YYcan be obtained viaXby means of a sequence of norm map additions
What follows next is the main result in all its glory and completeness, which basically tells us about the agreement of the poset structures we just saw. Thm4.2. *We have an order-reflecting and order-preserving bijection* rooted binary trees withn+2verticesNCpnAn+1
The remaining part of this result is the second bijection. The idea is to prove the correspondence between adding a norm map to the N-operad and a clockwise rotation operation to a tree. Here's a pictorial description of the part of the equivalence:

Poset Structures

Poset Structures The part's not as straightforward as this. I don't get into all the details; it just boils down to the following 3 cases:

Poset Structures from which the first and second cases are almost immediate, while the third needs a bit of work for whose details the reader's redirected to 1

. References ------ 1. Roitzheim, Constanze, Barnes, David, Balchin, Scott 2022 N-infinity operads and associahedra. Pacific Journal of Mathematics, 315 2. pp. 285-304. E-ISSN 0030-8730. doi:10.2140/pjm.2021.315.285 KARid:77048 link
https://kar.kent.ac.uk/77048/11/Roitzheim 2. Blumberg, Andrew J., and Michael A. Hill. "Operadic multiplications in equivariant spectra, norms, and transfers." Advances in Mathematics 285 2015: 658-708. link
https://pdf.sciencedirectassets.com/272585/1s2.0S0001870815X00144/1s2.0S000187081500256X/main.pdf?XAmzSecurityToken=IQoJb3JpZ2luX2VjEIL