Logarithmic morphisms, tangential basepoints, and little disks

TL;DR

This paper introduces the concept of virtual morphisms in logarithmic algebraic geometry to interpret topological maps algebraically, extending the theory of tangential basepoints and proving the formality of the little disks operad.

Contribution

It develops the theory of virtual morphisms, providing an algebro-geometric framework for tangential basepoints and operads in log schemes, and offers a new proof of the little disks operad's formality.

Findings

Virtual morphisms give a categorical framework for tangential basepoints.

The little disks operad is lifted to log schemes over integers.

A new algebraic proof of the little disks operad's formality is provided.

Abstract

We develop the theory of "virtual morphisms" in logarithmic algebraic geometry, introduced by Howell. It allows one to give algebro-geometric meaning to various useful maps of topological spaces that do not correspond to morphisms of (log) schemes in the classical sense, while retaining functoriality of key constructions. In particular, we explain how virtual morphisms provide a natural categorical home for Deligne's theory of tangential basepoints: the latter are simply the virtual morphisms from a point. We also extend Howell's results on the functoriality of Betti and de Rham cohomology. Using this framework, we lift the topological operad of little $2$-disks to an operad in log schemes over the integers, whose virtual points are isomorphism classes of stable marked curves of genus zero equipped with a tangential basepoint. The gluing of such curves along marked points is performed using virtual morphisms that transport tangential basepoints around the curves. This builds on Vaintrob's analogous construction for framed little disks, for which the classical notion of morphism in logarithmic geometry sufficed. In this way, we obtain a direct algebro-geometric proof of the formality of the little disks operad, following the strategy envisioned by Beilinson. Furthermore, Bar-Natan's parenthesized braids naturally appear as the fundamental groupoids of our moduli spaces, with all virtual basepoints defined over the integers.