Blog Post number 1

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$N_{\infty}$-operads and associahedra

June 16, 2023

Defining an $N_{\infty}$-operad using transfer systems

This time, we want a combinatorial understanding of $N_{\infty}$-operads in a suitable $G$-equivariant setting. The goal is establishing, for $G=C_{p^n}$, a one-one correspondence between the natural order on the collection of $N_{\infty}$-operads and the poset structure of Stasheff’s $(n+1)$-associahedron, $\mathcal{A}_{n+1}$. This is work by Balchin-Barnes-Roitzheim[1].

Instead of the traditional way, as mentioned right at the article’s outset, we want to understand $N_{\infty}$-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

$\mathbf{Thm} \ 1.1. \ $

\[\text{Ho}\left(N_{\infty}\text{-op} \right) \xrightarrow{\sim} \mathfrak{Ind}\]

where $N_{\infty}\text{-op}$ is the category of $N_{\infty}$-operads, and $\mathfrak{Ind}$ 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.

$\mathbf{Def} \ 1.2. \ $ A categorical coefficient system is a contravariant functor

\[\underline{\mathcal{C}} : \text{Orb}_{G} \rightarrow \mathcal{Cat}\]

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.

$\mathbf{Def} \ 1.3. \ $ Suppose $\underline{\mathcal{Set}}$ is the symmetric monoidal coefficient system of finite sets, with $\underline{\mathcal{Set}}(H) = \mathcal{Set}^{H}$. The symmetric monoidal operation is the disjoint union.

Now the idea is to associate to every $N_{\infty}$-operad a sub coefficient system of $\underline{\mathcal{Set}}$.

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

$\mathbf{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 $\mathcal{Z}$, 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 [].

$\mathbf{Lemma} \ 1.5. \ $ An indexing system determines (and is also determined by) a set $\mathcal{Z}_{H}$ for $H \leq G$, with subgroups of $H$ satisfying the following axioms:

  • $H/H \in \mathcal{Z}(H)$

  • $H/H’ \in \mathcal{Z}(H) \Rightarrow gHg^{-1}/gH’g^{-1} \in g\mathcal{Z}g^{-1}$

  • $H/H’ \in \mathcal{Z}(H) \ \wedge \ H’/K \in \mathcal{Z}(h’) \Rightarrow H/K \in \mathcal{Z}(H)$

  • $H/H’ \in \mathcal{Z}(H) \Rightarrow K(K \cap H’) \in \mathcal{K} \ , \forall \ K \leq H$

All this data is referred to as a transfer system.

Alternatively, we have the following definition.

$\mathbf{Def} \ 1.6. \ $ For a transfer system $\mathcal{Z}(H)$ 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 $N_{\infty}$-operad that I state next.

$\mathbf{Cor} \ 1.7. \ $

$G$ is a finite group. An $N_{\infty}$ operad for $G$, upto homotopy is defined as the data of the set $X = \{N_{H}^{K} \mid 1 \leq K < H \leq G \}$ satisfying the following (and all conjugates thereof)

  • $N_{H}^{K} \in X \ \wedge M < H \Rightarrow N_{K \cap M}^{M} \in X$ (Restriction Axiom aka $\mathfrak{RA}$)
  • $N_{L}^{K} \in X \ \wedge N_{H}^{K} \in X \Rightarrow N_{L}^{H} \in X$ (Composition Axiom aka $\mathfrak{CA}$)

It’s important to note that the notation choice makes sense so that the Mackey functor of an algebra $A$ over an $N_{\infty}$-operad $\mathcal{O}$ has multiplicative norm maps $N_{K}^{H}: \pi_{0}^{K}(A) \rightarrow \pi_{0}^{H}(A)$.

The case of $G= C_{p^n}$

We now look at the case of interest, i.e. when $G= C_{p^n}$ where $p$ is a prime.

The goal is to understand the sequence $\{ \mid N_{\infty}(C_{p^n}) \mid \}$ .

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 $\mathfrak{RA}$ and $\mathfrak{CA}$ to eliminate redundant structures.

The third structure i.e. $\{N_{0}^{1}, N_{1}^{2} \}$ isn’t an $N_{\infty}$-structure using $\mathfrak{CA}$ and the sixth and eighth structures i.e. $\{N_{0}^{2}, N_{1}^{2} \} \ , \ \{N_{0}^{2} \}$ aren’t $N_{\infty}$-structures by an application of $\mathfrak{RA}$.

Thus we can conclude that $\mid N_{\infty}(C_{p^2}) \mid = 5$

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.

$\mathbf{Def} \ 2.1. \ $ The Catalan numbers are terms of the sequence $\{ C_n \}$ 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.

\[C_0 = 1\] \[C_{n+1} = \sum_{l=0}^{n}C_l C_{n-l} \\ , \\ n\geq 0\]

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

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

BBR product

We now define an operation on $N_{\infty}$-operads that I call the Balchin-Barnes-Roitzheim (BBR) product, also called the o-dot product.

\[\odot: N_{\infty}(C_{p^k}) \times N_{\infty}(C_{p^{l}}) \rightarrow N_{\infty}(C_{p^{k+l+2}})\]

For $X \in N_{\infty}(C_{p^k})$ and $X’ \in N_{\infty}(C_{p^l})$, $X \odot X’ \in N_{\infty}(C_{p^{k+l+2}})$ 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 $\Sigma^{j}N_{a}^{b} = N_{a+j}^{b+j}$

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 $X,Y \in N_{\infty}(C_{p^2})$

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.

$\mathbf{Prop} \ 3.1. \ $ $X \in N_{\infty}(C_{p^k})$ and $Y \in N_{\infty}(C_{p^l})$. $X \odot Y$ is an object in $N_{\infty}(C_{p^{k+l+2}})$ for $k,l \geq -1$. Also conversely, if we’ve $X \odot Y \in N_{\infty}(C_{p^{k+l+2}})$ then $X$ and $Y$ are $N_{\infty}$ structures.

Now we make our next step towards computing $\mid N_{\infty}(C_{p^{n}}) \mid$

$\mathbf{Prop} \ 3.2. \ $ We have the following recurrence relation

\[|N_{\infty}(C_{p^{-1}}) | = 1\] \[|N_{\infty}(C_{p^n})| = \sum_{q=0}^{n}|N_{\infty}(C_{p^{q-1}})| |N_{\infty}(C_{p^{n-q-1}})|, \\ n \geq 0\]

The idea of the proof is to use the last proposition after showing that every $X \in N_{\infty}(C_{p^n})$ can be written as $X = Y \cdot Z$ where $Y \in N_{\infty}(C_{p^{q-1}})$ and $Z \in N_{\infty}(C_{p^{n-q-1}})$

Now we begin with $j \in \mathbb{Z}$ such that it’s the minimum integer such that $N_{n}^{j}$ is in $X$. The cases of $j=0$ and $j=n$ are straightforward, with $Y = \emptyset$ and $Z = \emptyset$ in both cases.

Now if $O < j < n$, then clearly we can split off $C_{p^i}$ with $0 \leq i < j$. Let’s call this part $Y \in N_{\infty}(C_{p^{j-1}})$ and the remaining part $X’$ Now it’s easy to see that this part’s like $\emptyset \cdot Z$ for some $Z \in N_{\infty}(C_{p^{n-j-1}})$.

We can thus conclude that $X = Y \cdot Z$

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

$\mathbf{Cor} \ 3.3. \ $

For $G= C_{p^n}$, and $X \in N_{\infty}(C_{p^n})$ we can write $X = Y \cdot Z$ for some suitable $N_{\infty}$-operads $Y$ and $Z$.

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

$\mathbf{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 $Y$ $$X < Y \Leftrightarrow \\{ \text{Y can be obtained from}\\ X \\ \text{by means of a finite sequence of rotations} \\}$$ $\mathbf{Def} \\ 4.1. \\ $ The poset structure on binary trees with $n+1$ vertices induced by the above relation is referred to as Stasheff's $n$-associahedron $\mathcal{A}_{n}$. We now equip $N_{\infty}$-operads for $G= C_{p^n}$ with a poset structure by defining the following order relation: For $X, Y \in N_{\infty}(C_{p^n})$ $$X < Y \Leftrightarrow \\{ Y \\ \text{can be obtained via}\\ X \\ \text{by means of a sequence of norm map additions} \\}$$ 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. $\mathbf{Thm} \\ 4.2. \\ $ *We have an order-reflecting and order-preserving bijection* $$\\{\text{rooted binary trees with} \\ n+2 \\ \text{vertices} \\} \Leftrightarrow \\{N_{C_{p^n}} \\} \Leftrightarrow \mathcal{A}_{n+1}$$ The remaining part of this result is the second bijection. The idea is to prove the correspondence between adding a norm map to the $N_{\infty}$-operad and a clockwise rotation operation to a tree. Here's a pictorial description of the $\Leftarrow$ part of the equivalence:

Poset Structures $$\Updownarrow$$

Poset Structures The $\Rightarrow$ 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) (KAR id:77048) [link]( 2. Blumberg, Andrew J., and Michael A. Hill. "Operadic multiplications in equivariant spectra, norms, and transfers." Advances in Mathematics 285 (2015): 658-708. [link](