[Paper Review] Notes on Restricted Inverse Limits of Categories

Inna Entova-Aizenbud|arXiv (Cornell University)|Apr 5, 2015

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper introduces the concept of a restricted inverse limit of categories, focusing on the category of polynomial representations of $GL_\infty = \bigcup_{n\geq 0} GL_n$, which is shown to be the restricted inverse limit of the categories of polynomial representations of $GL_n$. The key result is an equivalence of symmetric monoidal abelian categories between $Rep({\mathfrak{gl}}_\infty)_{poly}$ and the restricted inverse limit $\varprojlim_{n\in{\mathbb{Z}}_+,\text{restr}}Rep({\mathfrak{gl}}_n)_{poly}$, established via a compatible system of restriction functors and a filtration-based construction.

ABSTRACT

We describe the framework for the notion of a restricted inverse limit of categories, with the main motivating example being the category of polynomial representations of the group $GL_{\infty}$. This category is also known as the category of strict polynomial functors of finite degree, and it is the restricted inverse limit of the categories of polynomial representations of $GL_n$. This note is meant to serve as a reference for future work.

Motivation & Objective

• To formalize the notion of a restricted inverse limit of categories, particularly in the context of infinite-dimensional Lie groups and their representations.
• To establish that the category of polynomial representations of $GL_\infty$ arises as a restricted inverse limit of the categories of polynomial representations of $GL_n$.
• To demonstrate that the restriction functors $\mathfrak{Res}_{n-1,n}$ are compatible with ${\mathbb{Z}}_+$-filtrations and satisfy the conditions for a restricted inverse limit.
• To prove an equivalence between $Rep({\mathfrak{gl}}_\infty)_{poly}$ and the restricted inverse limit category, preserving symmetric monoidal and abelian structures.

Proposed method

• Define the restricted inverse limit of a sequence of categories $\mathcal{C}_n$ and functors $\mathcal{F}_{n-1,n}: \mathcal{C}_n \to \mathcal{C}_{n-1}$ as the category of coherent compatible families of objects and isomorphisms.
• Equip each category $Rep({\mathfrak{gl}}_n)_{poly}$ with a ${\mathbb{Z}}_+$-filtration based on the length of the corresponding partition.
• Show that the restriction functors $\mathfrak{Res}_{n-1,n}$ are shortening functors and preserve the filtration structure.
• Prove that the inverse limit of the ${\mathbb{Z}}_+$-filtered categories $Rep({\mathfrak{gl}}_n)_{poly}$ satisfies the conditions of a restricted inverse limit via Proposition 6.1.1.
• Construct a symmetric monoidal equivalence $\Gamma_{\text{lim}}: Rep({\mathfrak{gl}}_\infty)_{poly} \to \varprojlim_{n\in{\mathbb{Z}}_+,\text{restr}}Rep({\mathfrak{gl}}_n)_{poly}$ using the functors $\Gamma_n = (\cdot)^{{\mathfrak{gl}}_n^\perp}$.
• Define the inverse functor $\Gamma_{\text{lim}}^*$ by taking the direct limit of the compatible system of modules $M_n$ along the isomorphisms $\phi_{n-1,n}$, yielding a $\mathfrak{gl}_\infty$-module structure.

Experimental results

Research questions

• RQ1 How can one define a restricted inverse limit of categories in a way that preserves categorical structures such as abelian, additive, and symmetric monoidal properties?
• RQ2 What conditions ensure that the inverse limit of a system of categories with filtered functors is equivalent to the category of polynomial representations of $GL_\infty$?
• RQ3 How do the restriction functors $\mathfrak{Res}_{n-1,n}$ interact with the ${\mathbb{Z}}_+$-filtration on $Rep({\mathfrak{gl}}_n)_{poly}$?
• RQ4 Is the category $Rep({\mathfrak{gl}}_\infty)_{poly}$ naturally equivalent to the restricted inverse limit of the $Rep({\mathfrak{gl}}_n)_{poly}$ categories?
• RQ5 Can the inverse limit construction be made compatible with the symmetric monoidal structure of the representation categories?

Key findings

• The category $Rep({\mathfrak{gl}}_\infty)_{poly}$ is equivalent to the restricted inverse limit $\varprojlim_{n\in{\mathbb{Z}}_+,\text{restr}}Rep({\mathfrak{gl}}_n)_{poly}$ via a symmetric monoidal equivalence $\Gamma_{\text{lim}}$.
• The restriction functors $\mathfrak{Res}_{n-1,n}$ are shortening functors and preserve the ${\mathbb{Z}}_+$-filtration, enabling the construction of the restricted inverse limit.
• The inverse limit category inherits the abelian, additive, and symmetric monoidal structures from the individual $Rep({\mathfrak{gl}}_n)_{poly}$ categories.
• The inverse functor $\Gamma_{\text{lim}}^*$ is constructed by taking the direct limit of the compatible system $M_n$ along the isomorphisms $\phi_{n-1,n}$, yielding a $\mathfrak{gl}_\infty$-module structure.
• The composition $\Gamma_{\text{lim}}^* \circ \Gamma_{\text{lim}}$ is naturally isomorphic to the identity functor on $Rep({\mathfrak{gl}}_\infty)_{poly}$, confirming the equivalence.
• The construction is equivalent to the stable inverse limit defined in [3], confirming consistency with prior work in the field.