Towards functor exponentiation

We consider a possible framework to categorify the exponential map exp⁡(−f) given the categorification of a generator f of sl2 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.