ninfty: A software package for homotopical combinatorics

TL;DR

n_infty provides a computational framework for enumerating $G$-transfer systems that encode equivariant multiplicativity in homotopy theory, linking the poset of transfer systems with the homotopy type of $N_\infty$-operads via $\mathsf{Tr}(G) \cong \mathrm{Ho}(N_\infty(G))$. It implements Rubin's closure algorithm in a header-only C++20 library, with Sage-based data generation for finite groups and arbitrary lattices, enabling scalable exploration of invariants and structures in $N_\infty(G)$. The package exposes a rich set of counting statistics (e.g., numbers of transfer systems, cosaturated/saturated systems, premodeled structures, Quillen model structures) and export options (TikZ diagrams, data sheets, Sage-ready formats), while candidly acknowledging memory constraints that limit large-scale computations. By combining high-performance enumeration with data-generation workflows and visualization facilities, ninfty provides a practical bridge between the theory of equivariant operads and computational experimentation, accelerating discovery in homotopical combinatorics.

Abstract

We introduce ninfty, a header-only C++ library distributed under an MIT Open Source License designed for the study of enumeration problems arising in homotopical combinatorics. The ninfty repository moreover contains a folder with data files for many common finite groups. This is in addition to Sage code which can be used to generate input data for further finite groups, and Sage code for generating input data for abstract lattices which may not arise at the subgroup lattice of a group.

Figures (5)

• Figure 1: The 68 transfer systems for $Q_8$.
• Figure :
• Figure :
• Figure :
• Figure :

Theorems & Definitions (1)

• Definition 1