PlanetPhysics/Grassmann Hopf Algebroid Categories and Grassmann Categories

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Grassmann-Hopf Algebroid Categories and Grassmann Categories

The categories whose objects are either Grassmann-Hopf al/gebras , or in general $G-H$ algebroids, and whose morphisms are $G-H$ homomorphisms are called Grassmann-Hopf Algebroid Categories .

Although carrying a similar name, a quite different type of Grassmann categories have been introduced previously:

Grassmann Categories (as in ) are defined on $k$ letters over nontrivial abelian categories Failed to parse (unknown function "\A"): {\displaystyle \mathbf{\A}} as full subcategories of the categories Failed to parse (unknown function "\A"): {\displaystyle F_{\mathbf{\A}}(x_1,...,x_k)} consisting of diagrams satisfying the relations: $x_{i}x_{j}+x_{j}x_{i}=0$ and $x_{i}x_{i}=0$ with additional conditions on coadjoints, coproducts and morphisms.

They were shown to be equivalent to the category of right modules over the endomorphism ring of the coadjoint $S(R)$ which is isomorphic to the Grassmann--or exterior--ring over $R$ on $k$ letters $E_{R}(X_{1},...,X_{N})$.