Lie supergroups vs. super Harish-Chandra pairs: a new equivalence

It is known that there exists a natural functor $\mathrm{\Phi }$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by other authors, to different levels of generality, but always elaborating on Koszul’s original idea. In this paper, I provide two new backwards equivalences, i.e. two different functors ${\mathrm{\Psi }}^{\circ }$ and ${\mathrm{\Psi }}^e$ that construct a Lie supergroup (thought of as a special group-valued functor) out of a given super Harish-Chandra pair, so that any Lie supergroup is recovered from its naturally associated super Harish-Chandra pair; more precisely, both ${\mathrm{\Psi }}^{\circ }$ and ${\mathrm{\Psi }}^e$ are quasi-inverse to the functor $\mathrm{\Phi }$.