Higher configuration operads by way of quiver Grassmannians

TL;DR

This paper constructs new operads from quiver Grassmannians associated with finite dimensional vector spaces, linking them to moduli spaces of curves and trees, and providing a general framework for operad construction from functors.

Contribution

It introduces a novel construction of operads using quiver Grassmannians and develops a general method to build operads from suitable functors in category theory.

Findings

Operads contain moduli spaces of marked rational curves and trees of projective spaces as suboperads.

Operads can be realized as disjoint unions of quiver Grassmannians.

A general framework for constructing operads from functors satisfying certain conditions.

Abstract

We introduce a construction that associates, to each finite dimensional k-vector space V, a family of projective k-varieties that comes equipped with the structure of a operad in the category of k-schemes. When dim V = 1, this operad contains, as a suboperad, the family of moduli spaces of stably marked rational curves. For general V, our operad contains the family of Chen-Gibney-Krashen moduli spaces of stably marked trees of projective spaces as a suboperad. We realize our operad as part of a larger theory that describes how to construct operads from suitable functors. Given a category C that satisfies conditions allowing us to consider the operation of "substituting a value into an argument" within C, and given any functor F on C satisfying a variant of right-exactness, we construct a version of a set-valued operad whose inputs are given by objects in C. When C happens to be the category of nonempty finite sets, we obtain operads in the classic sense. Finally, we show that in many cases it is possible to realize these operads as disjoint unions of quiver Grassmannians.