PlanetPhysics/Superalgebroids in Higher Dimensions

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

Double algebroids

A double $R$--algebroid  consists of a

double category $D$, as detailed in ref., such that each [[../IndexOfCategories/|category structure]] has the additional structure of an $R$--algebroid. More precisely, a double [[../RAlgebroid/|$R$--algebroid]] Failed to parse (unknown function "\D"): {\displaystyle \D} involves four related [[../RAlgebroid/|$R$--algebroids]]:

Failed to parse (unknown function "\del"): {\displaystyle (D,D_1,\del^0_1 ,\del^1_1 , \vep_1 , +_1 , \circ _1 , ._1) ,\qquad &(D,D_2,\del^0_2 , \del ^1_2 , \vep_2 , +_2 , \circ _2 , ._2 )\\ (D_1,D_0, \delta^0_1 ,\delta^1_1 , \vep , + , \circ , .) ,\qquad &(D_2 , D_0 , \delta^0_2 , \delta^1_2 , \vep , + , \circ , .) } that satisfy the following rules:

 \item[i)] Failed to parse (unknown function "\del"): {\displaystyle \delta^i_2 \del^j_2 = \delta ^j_1 \del ^i_1}
 for </math>i,j \in \{0,1\}Failed to parse (unknown function "\med"): {\displaystyle   \med \item[ii)] <math>  \del^i_2 ( \a +_1 \be) = \del^i_2\a +\del^i_2\be , \qquad &\del^i_1 ( \a +_2 \be) = \del^i_1 \a +\del^i_1\be \\ \del^i_2 ( \a \circ _1 \be) = \del^i_2 \a \circ \del^i_2 \be ,\qquad & \del^i_1 (\a \circ _2 \be ) = \del^i_1\a \circ \del^i_1\be  }
 for Failed to parse (unknown function "\a"): {\displaystyle i = 0,1 , \a,\be \in D}
 and both sides are defined.  \med \item[iii)] Failed to parse (unknown function "\a"): {\displaystyle   r ._1 (\a+_2 \be) = (r ._1 \a) +_2 (r ._1\be ) ,\qquad & r ._2 (\a +_1 \be ) = (r ._2 \a ) +_1 (r ._2\be )\\ r ._1 (\a \circ _2 \be ) = (r ._1\a ) \circ _2 (r ._1 \be ) , \qquad & r ._2 (\a \circ _1 \be ) = (r ._2 \a ) \circ _1 (r ._2\be )\\ r ._1 ( s ._2 \be ) &= s ._2 ( r._1 \be )  }
 for all Failed to parse (unknown function "\a"): {\displaystyle \a ,\be \in D, ~r,s \in R}
 and both sides are defined.  \med \item[iv)] Failed to parse (unknown function "\a"): {\displaystyle   (\a +_1 \be ) +_2 (\gamma +_1 \lambda )& = (\a +_2 \gamma ) +_1 (\be +_2 \lambda ) ,\\ (\a \circ _1 \be ) \circ _2 (\gamma \circ _1 \lambda )& = (\a \circ _2 \gamma ) \circ _1 (\be \circ _2 \lambda )\\ (\a +_i \be ) \circ _j (\gamma +_i \lambda ) &= (\a \circ _j \gamma ) +_i (\be \circ _j \lambda )  }
 for $i\ne j$, whenever both sides are defined.

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

Category of double algebroids

A morphism Failed to parse (unknown function "\D"): {\displaystyle f : \D \to \E} of double algebroids is then defined as a [[../TrivialGroupoid/|morphism]] of truncated cubical sets which [[../Commutator/|commutes]] with all the algebroid structures. Thus, one can construct a category $𝐃𝐀$ of double algebroids and their morphisms. The main construction in this subsection is that of two functors $\eta ,{\eta }^{\prime }$ from this category $𝐃𝐀$ to the category $𝐂𝐌$ of crossed modules of algebroids.

Let $D$ be a double algebroid. One can associate to $D$ a crossed module Failed to parse (unknown function "\lra"): {\displaystyle \mu : M \lra {D}_1} . Here $M(x,y)$ will consist of elements $m$ of $D$ with [[../PiecewiseLinear/|boundary]] of the form: 0 1 Failed to parse (unknown function "\del"): {\displaystyle \del m = \quadr{a}{1_y}{1_x}{ 0_{xy}}~, } that is </math>M(x,y) = \{ m \in D : \del^1_1 m = 0_{xy} , \del^0_2 m = 1_x,\del^1_2 m = 1_y \}$.===CubicandHigherdimensionalalgebroids===Onecanextendtheabovenotionofdoublealgebroidtocubicandhigherdimensionalalgebroids.The[[../PreciseIdea/|concepts]]of2-algebroid,3-algebroid,...,<math>n$--algebroid and superalgebroid are however quite distinct from those of double, cubic,..., n--tuple algebroid, and have technically less complicated definitions.