Towards functor exponentiation

TL;DR

This paper proposes a framework for categorifying the exponential map in the context of categorified Lie algebra generators, using diagrammatic relations and complexes, though it remains an approximation due to non-invertibility of certain functors.

Contribution

It introduces a novel diagrammatic approach to categorify the exponential map related to rak{sl}_2, extending Lauda's categorification of generators.

Findings

Taylor expansions become complexes of categorified divided powers

Hom spaces are described by diagrammatic relations combining nilHecke algebra and short strand generators

The framework is an approximation due to non-invertible functors

Abstract

We consider a possible framework to categorify the exponential map exp(-f) given the categorification of a generator f of $\frak{sl}_2$ by Lauda. In this setup the Taylor expansions of exp(-f) and exp(f) turn into complexes built out of categorified divided powers of f. Hom spaces between tensor powers of categorified f are given by diagrammatics combining nilHecke algebra relations with those for a additional "short strand" generator. The proposed framework is only an approximation to categorification of exponentiation, because the functors categorifying exp(f) and exp(-f) are not invertible.