Foundations of Differential Calculus for modules over posets

TL;DR

The paper develops a calculus for $k\mathcal{C}$-modules by categorifying gradient and divergence through line digraphs and Kan extensions, enabling local analysis of modules over categories with a poset structure. It defines a categorical gradient $\nabla$ and left/right divergences $\nabla^*,\nabla_*$, constructs corresponding Laplacians, and studies their interaction via Hom and Euler pairings, with a focus on finite posets. A central result provides a necessary and sufficient condition for the gradient to vanish on subposets generated by line-connected trees, and the work clarifies how gradient information interacts with the rank invariant and module restrictions. The framework is illustrated through examples on commutative ladders and grid posets, revealing how local calculus can yield structural insight even for representations that are globally intractable due to wild representation type. The methodology aims to underpin applications to generalized persistence modules and related topological data analysis tasks as a local, computable invariant theory for $k\mathcal{C}$-modules.

Abstract

Let $k$ be a field and let $C$ be a small category. A $k$-linear representation of $C$, or a $kC$-module, is a functor from $C$ to the category of finite dimensional vector spaces over $k$. Unsurprisingly, it turns out that when the category $C$ is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. This paper offers a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of $kC$-modules, under some restrictions on the category $C$. As a starting point, for a $kC$-module $M$ we define its gradient \emph{gradient} $\nabla[M]$ as a virtual module in the appropriate Grothendieck group. Pushing the analogy with ordinary differential calculus and discrete calculus on graphs, we define left and right divergence via the appropriate left and right Kan extensions and two bilinear pairings on modules and study their properties, specifically with respect to adjointness relations between the gradient and the left and right divergence. The left and right divergence are shown to be rather easily computable in favourable cases. Having set the scene, we concentrate specifically on the case where the category $C$ is a finite poset. Our main result is a necessary and sufficient condition for the gradient of a module $M$ to vanish under certain hypotheses on the poset. We next investigate implications for two modules whose gradients are equal. Finally we consider the resulting left and right Laplacians, namely the compositions of the divergence with the gradient, and study an example of the relationship between the vanishing of the Laplacians and the gradient.

Figures (3)

• Figure 1: A quiver (left) and its line digraph (right) with the line components and their associated line digraphs in red and blue.
• Figure 2: Left: Four sample posets given by their respective Hasse diagrams. Centre and Right: For an arbitrary module $M\in k\mathcal{P}\hbox{-}\EuR{mod}$, the corresponding modules $\phi^*M$ and $\beta^*M$ in $k\widehat{\mathcal{P}}\hbox{-}\EuR{mod}$. The gradient is given by the formal difference $[\phi^*M]-[\beta^*M]$ in $\EuR{Gr}(k\widehat{\mathcal{P}})$
• Figure 3: Example dimension vectors of a $k\mathcal{A}^{e,\ast}_{10,10}$-module (left) and its gradient (right). Illustrations show the Hasse diagrams of the posets $\mathcal{A}_{10,10}$ and $\widehat{\mathcal{A}}_{10,10}$, respectively.

Theorems & Definitions (102)

• Theorem 1: Thm. \ref{['Thm:grad0']}
• Theorem 2: Thm. \ref{['Thm:Rank']}
• Theorem 3: Thm. \ref{['Thm:Divergence-tree']}
• Proposition 4: Prop. \ref{['Prop:Lap-Grad-Kernels']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Mitchell
• Definition 2.4
• Lemma 2.5
• proof