Abstract: We start by considering the notions of a torsor and a gerbe for a discrete Abelian group A (over a discrete set X). We describe central extensions of a group G by A as multiplicative A-torsors on X=G. A categorification of this construction is given by the notion of a gerbal central extension of a group G by an Abelian group A. We classify gerbal A-central extensions of the group G by A-valued 3-cocycles of G.

Then we recall the construction of determinantal gerbe Det(V) of an infinite dimensional Tate vector space V (e.g. V=k((s)) ) due to Kapranov. We explain that the obstruction of Det(V) to be GL(V)-equivariant provides the well known central extension of GL(V) by k^*. Given a 2-Tate vector space V, e.g. V=k((s))((t)), we consider the determinantal 2-gerbe 2-Det(V). The obstruction of 2-Det(V) to be GL(V)-equivariant provides a gerbal central extension of GL(V) by k^*. This construction leads to gerbal central extensions of double loop groups.

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