# PlanetPhysics/Grassmann Hopf Algebroid Categories and Grassmann Categories

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### Grassmann-Hopf Algebroid Categories and Grassmann CategoriesEdit

The categories whose objects are either *Grassmann-Hopf al/gebras* , or in general algebroids,
and whose morphisms are * 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 ^{[1]}) are defined *on 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: and 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 which is isomorphic to the Grassmann--or exterior--ring over on letters .

## All SourcesEdit

^{[1]}^{[2]}^{[3]}^{[4]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}Barry Mitchell.*Theory of Categories*., Academic Press: New York and London.(1965), pp. 220-221. - ↑
B. Fauser:
*A treatise on quantum Clifford Algebras*. Konstanz, Habilitationsschrift. (PDF at arXiv.math.QA/0202059).(2002). - ↑
B. Fauser: Grade Free product Formulae from Grassmann--Hopf Gebras., Ch. 18 in R. Ablamowicz, Ed.,
*Clifford Algebras: Applications to Mathematics, Physics and Engineering*, Birkh\"{a}user: Boston, Basel and Berlin, (2004). - ↑
I.C. Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non-Abelian Algebraic Topology.
*in preparation*, (2008).