This work is a companion to our article "Super Kac-Moody 2categories," which introduces super analogs of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier. In the case of sl 2 , the super Kac-Moody 2-category was constructed already in [A. Ellis and A. Lauda, "An odd categorification of Uq(sl 2 )"], but we found that the formalism adopted there became too cumbersome in the general case. Instead, it is better to work with 2-supercategories (roughly, 2-categories enriched in vector superspaces). Then the Ellis-Lauda 2-category, which we call here a Π-2-category (roughly, a 2category equipped with a distinguished involution in its Drinfeld center), can be recovered by taking the superadditive envelope then passing to the underlying 2-category. The main goal of this article is to develop this language and the related formal constructions, in the hope that these foundations may prove useful in other contexts.

We introduce an odd version of the nilHecke algebra and develop an odd analogue of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogues of the cohomology rings of Grassmannians. Like their even counterparts, odd nilHecke algebras categorify the positive half of quantum sl(2).

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q = −1 case leads to a -graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting we describe counterparts of the elementary and complete symmetric functions, power sums, Schur functions, and combinatorial interpretations of associated change of basis relations.

ABSTRACT. We define a 2-category that categorifies the covering Kac-Moody algebra for sl 2 introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a × 2 -grading giving its Grothendieck group the structure of a free module over the group algebra of × 2 . By specializing the 2 -action to +1 or to −1, the construction specializes to an "odd" categorification of sl 2 and to a supercategorification of osp 1|2 , respectively.

Abstract. We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.