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%%% Primary Title: Grassmann-Hopf algebroid categories and Grassmann categories
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%%% Filename: GrassmannHopfAlgebroidCategoriesAndGrassmannCategories.tex
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%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\subsection{Grassmann-Hopf Algebroid Categories and Grassmann Categories}
\begin{definition}
The \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are either \emph{Grassmann-Hopf al/gebras}, or in general $G-H$ \htmladdnormallink{algebroids}{http://planetphysics.us/encyclopedia/Algebroids.html},
and whose \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are \emph{$G-H$ \htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}} are called \emph{Grassmann-Hopf Algebroid Categories}.
\end{definition}
Although carrying a similar name, a quite different \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of Grassmann categories have been
introduced previously:
\begin{definition}
\emph{Grassmann Categories} (as in \cite{BM65}) are defined \emph{on $k$ letters over nontrivial \htmladdnormallink{abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory2.html}} $\mathbf{\A}$ as \emph{full subcategories} of the categories $F_{\mathbf{\A}}(x_1,...,x_k)$ consisting of \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} satisfying the \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html}: $x_i x_j + x_j x_i = 0$ and $ x_i x_i = 0 $ with additional conditions on coadjoints, \htmladdnormallink{coproducts}{http://planetphysics.us/encyclopedia/CompactProjectiveGenerator.html} and morphisms.
They were shown to be equivalent to the category of right \htmladdnormallink{modules}{http://planetphysics.us/encyclopedia/RModule.html} over the endomorphism ring of the
coadjoint $S(R)$ which is isomorphic to the Grassmann--or exterior--ring over $R$ on $k$ letters $E_R(X_1,..., X_N)$.
\end{definition}
\begin{thebibliography}{9}
\bibitem{BM65}
Barry Mitchell.{\em Theory of Categories}., Academic Press: New York and London.(1965), pp. 220-221.
\bibitem{Fauser2002}
B. Fauser: \emph{A treatise on quantum Clifford Algebras}. Konstanz, Habilitationsschrift. (PDF at arXiv.math.QA/0202059).(2002).
\bibitem{Fauser2004}
B. Fauser: Grade Free product Formulae from Grassmann--Hopf Gebras., Ch. 18 in R. Ablamowicz, Ed.,
\emph{Clifford Algebras: Applications to Mathematics, Physics and Engineering}, Birkh\"{a}user: Boston, Basel and Berlin, (2004).
\bibitem{BBGGk8}
I.C. Baianu, R. Brown J.F. Glazebrook, and G. Georgescu, Towards Quantum Non-Abelian Algebraic Topology. \emph{in preparation}, (2008).
\end{thebibliography}
\end{document}
