Categorified Möbius Inversion
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There are many ways to imagine categorifying Möbius functions on posets, and Möbius inversion.
[This paper](https://arxiv.org/abs/1402.2131) gives quite a good survey of the methods
which have previously been explored.
For my research, I find that I require a homological categorification.
By this, I mean replacing the incidence algebra on a locally finite poset $P$ by
a category of functors from intervals of $P$ to the homotopy category $K(\mathcal{A})$
of bounded chain complexes over an abelian category $\mathcal{A}$. If you'd like, just
take $\mc{A}$ to be the category of abelian groups and homomorphisms,
or a more general module category if that would be preferable.
Note: The ideas here do not turn out to quite give the machinery I am looking for, which
is why I am publishing it on my blog as opposed to on the arXiv.
However, I hope this post will eventually lead to a proper article.
Please share any thoughts you might have or ideas to improve the construction.
**Definition**
Let $P$ be a locally finite poset, and let $\Int(P)$ denote the poset of intervals of $P$ partially ordered by inclusion.
Let $\mc{A}$ be a monoidal abelian category.
The *incidence category* of $P$ is the functor category $$\mc{I}(P)=[\Int(P),K(\mc{A})]$$
whose objects are functors from $\Int(P)$ to $K(\mc{A})$ and whose morphisms are natural transformations.
We will use the shorthand notation $F([a,b])=F(a,b)$.
Next we define a monoidal structure on $\mc{I}(P)$ which categorifies the multiplication
in the incidence algebra. Note for any functor $F:P\to \mc{A}$, where $\mc{A}$ is monoidal abelian,
we can always define the *dual functor* $F^*:P^*\to\mc{C}$ via $F^*(x)=\Hom(F(x),1)$ where $1$ is
the monoidal identity in $\mc{A}$. For $a\leq b$ in $P$, define $$F^*(b\leq a):\Hom(F(b),1)\to\Hom(F(a),1)$$
by sending $g:F(b)\to 1$ to $g\circ F(a\leq b):F(a)\to 1$.
**Definition**
Define the *convolution* functor $*:\mc{I}(P)\times\mc{I}(P)\to\mc{I}(P)$ via
$$(F*G)(a,b)=\bigoplus_{z\in[a,b]}F(a,z)\otimes G^*(z,b)$$
with differential \ACcom{maybe this should use the differentials in $F(a,z)$ and $G(z,b)$
instead of the functors action on morphisms, and the last part uses functors action on morphisms}
$$d=\sum_{a\leq z\lessdot w\leq b}F\bigg([a,z]\subseteq[a,w]\bigg)\otimes G^*\bigg([z,b]\supseteq[w,b]\bigg)$$
and
$(F*G)\bigg([a,b]\subseteq [c,d]\bigg)$ is determined by its matrix elements $(F*G)\bigg([a,b]\subseteq [c,d]\bigg)_{z,w}$ for $z\in[a,b],w\in[c,d]$, where
$$(F*G)\bigg([a,b]\subseteq [c,d]\bigg)_{z,w}=\delta_{z\leq w}F([a,z]\subseteq[c,w])\otimes G([w,b]\subseteq[z,d]))$$
where $\delta_{z\leq w}$ is $1$ if $z\leq w$ and $0$ otherwise.
Given a morphism $(\eta_1,\eta_2):(F_1,G_1)\to (F_2,G_2)$,
where $\eta_1:F_1\to G_1,\eta_2:F_2\to G_2$ are natural transformations on functors
$F_1,F_2:\Int(P)\to K(\mc{A})$, $G_1,G_2:\Int(P^*)\to K(\mc{A})$,
define $\eta_1 *\eta_2:F_1*G_1\to F_2*G_2$ as follows. Given $x\in P$ define
$$(\eta_1*\eta_2)_x=(\eta_1)_x\otimes(\eta_2)_x.$$
To show that $\eta_1*\eta_2$ is a natural transformation, we must show the following diagram commutes
\begin{tikzcd}
(F_1*G_1)(X\otimes Y)\arrow[r,"f"]\arrow[d]&F_2*G_2\arrow[d]\\
C\arrow[r]&D
\end{tikzcd}
**Lemma**
Suppose that the functors $F$ and $G$ assign complexes with trivial differential.
Then the convolution $F*G$ is well defined.
**Example**
Given any functor $F\in \mc{I}(P)$, applying homology $H^*$ gives a new functor $H^*F\in\mc{I}(P)$,
where $(H^*F)(a,b)=H^*(F(a,b))$ is interpreted as having all zero differentials.
\ACcom{since we are in the homotopy category, do we have $F=H^*F$? Or do we need to pass to the derived category?}
**Example**
Suppose $F\in\Funct(P,\mc{A})$. Define $\hat{F}\in\mc{I}(P)$ via $\hat{F}(a,b)=C^*_{\sheaf}(F|_{[a,b]})$,
where the action on inclusions of intervals is the induced one.
**Example**
Define the functor $1_{\mc{A}}^P=1$ sending each $x\in P$ to the monoidal identity
$1_{\mc{A}}$ and for any $x\leq y\in P$, $1(x\leq y)$ is the identity map on $1_{\mc{A}}$.
Then we get a functor $\hat{1}=\hat{1}_{\mc{A}}^P\in\mc{I}(P)$.
**Example**
Define the *delta functor* $\Delta:\Int(P)\to K(\mc{A})$ via $\Delta(a,a)=1_{\mc{A}}$ $\Delta(a,b)=0_{\mc{A}}$
if $a\neq b$.
**Example**
Define the *Zeta functor* $Z:\Int(P)\to K(\mc{A})$ sending each nonempty interval $[a,b]$ to $1_{\mc{A}}$
and each inclusion of intervals to the identity map on $1_{\mc{A}}$. Notice that $Z=1_{\mc{A}}^{\Int(P)}$.
**Definition**
A *Möbius functor* for $P$ (with respect to $\mc{A}$) is a functor $M\in\mc{I}(P)$ such that
$$M*Z=D=Z*M.$$
Recall for a poset $P$ and $a,b\in P$, the Möbius function $\mu$ satisfies that $\mu(a,b)$ is equal to the
reduced Euler characteristic of the order complex $\Delta(a,b)$, the simplicial complex whose elements are
subsets $S\subseteq (a,b)$ which are totally ordered in $P$. This suggests the categorified version $M$ of
$\mu$ should satisfy $M(a,b)=H^*(\Delta(a,b))$ or even associate to $(a,b)$ the poset homology complex
$C^*((a,b),1)$ where $1$ is the functor $1:(a,b)\to\mc{A}$ sending each element to the monoidal identity
and all arrows to identity maps. In the case that $P$ is thin, the complex $C^*((a,b),1)$ is much simpler than
the chain complex associated to the order complex, but is homotopy equivalent in the case of $\Zmod$
(derived equivalent in general?).
**Definition** Define the contravariant functor $M:\Int(P)\to K(\mc{A})$ as follows:
$M(a,a)=\Z_{(0)}$, $M(a,b)=\Z_{(1)}$ for $a\lessdot b$, and
$M(a,b)=H^*(\Delta(a,b))$ for $a
Alex Chandler
Krener Assistant Professor
My research interests include machine learning, algebraic combinatorics, categorification, graph theory, knot theory, low dimensional topology, topological combinatorics.