PlanetMath: superalgebroids and higher dimensional algebroids

Definition 0.1 A double $R$ -algebroid consists of a double category $D$ , as detailed in ref.[2], such that each category structure has the additional structure of an $R$ -algebroid. More precisely, a double $R$ -algebroid $\mathsf{D}$ involves four related $R$ -algebroids:

$\begin{equation*}\begin{aligned}(D,D_1,\partial ^0_1 ,\partial ^1_1 , \varepsilo... ...lta^0_2 , \delta^1_2 , \varepsilon , + , \circ , .) \end{aligned}\end{equation*}$

that satisfy the following rules:

i)

$\delta^i_2 \del^j_2 = \delta ^j_1 \del ^i_1$ for $i,j \in \{0,1\}$

ii)

$\begin{equation*}\begin{aligned}\partial ^i_2 ( \alpha +_1 \beta ) = \partial ^i... ...ta ) = \partial ^i_1\alpha \circ \partial ^i_1\beta \end{aligned}\end{equation*}$

for $i = 0,1 , \a,\be \in D$ and both sides are defined.

iii)

$\begin{equation*}\begin{aligned}r ._1 (\alpha +_2 \beta ) = (r ._1 \alpha ) +_2 ... ...a )\\ r ._1 ( s ._2 \beta ) &= s ._2 ( r._1 \beta ) \end{aligned}\end{equation*}$

for all $\a ,\be \in D, ~r,s \in R$ and both sides are defined.

iv)

$\begin{equation*}\begin{aligned}(\alpha +_1 \beta ) +_2 (\gamma +_1 \lambda )& =... ...pha \circ _j \gamma ) +_i (\beta \circ _j \lambda ) \end{aligned}\end{equation*}$

for $i \neq j$ , whenever both sides are defined.

The definition of a double algebroid specified above was introduced by Brown and Mosa [1]. Two functors can be then constructed, one from the category of double algebroids to the category of crossed modules of algebroids, whereas the reverse functor is the unique adjoint (up to natural equivalence). The construction of such functors requires the following definition.

Category of Double Algebroids

A morphism $f : \mathsf{D}\to \mathcal E$ of double algebroids is then defined as a morphism of truncated cubical sets which commutes with all the algebroid structures. Thus, one can construct a category $\mathbf{DA}$ of double algebroids and their morphisms. The main construction in this subsection is that of two functors $\eta,\eta'$ from this category $\mathbf{DA}$ to the category $\mathbf{CM}$ of crossed modules of algebroids.

Let ${D}$ be a double algebroid. One can associate to ${D}$ a crossed module $\mu : M \lra {D}_1$ . Here $M(x,y)$ will consist of elements $m$ of ${D}$ with boundary of the form: 0 1 \begin{equation} \del m = \quadr{a}{1_y}{1_x}{ 0_{xy}}~, \end{equation}that is $M(x,y) = \{ m \in D : \del^1_1 m = 0_{xy} , \del^0_2 m = 1_x,\del^1_2 m = 1_y \}$ .

Cubic and Higher dimensional algebroids

One can extend the above notion of double algebroid to cubic and higher dimensional algebroids. The concepts of 2-algebroid, 3-algebroid,..., $n$ -algebroid and superalgebroid are however quite distinct from those of double, cubic,..., n-tuple algebroid, and have technically less complicated definitions.

Definitions of double, and higher dimensional, algebroids, superalgebroids and generalized superalgebras

Category of Double Algebroids

Cubic and Higher dimensional algebroids

Bibliography