Comparison of the Algebras

Introduction

This article compares the eight algebras $\mathbb{R}, \mathbb{C}, \mathbb{D}, \mathbb{D}', \mathbb{H}, \mathbb{H}_{\mathrm{s}}, \mathbb{B}, \mathbb{H}_{\mathbb{D}}$ as algebras. It states, for each invariant of the algebra structure, the situation of the eight, in tables with the eight algebras as columns in the fixed order of The Eight Algebras Compared, and explains what each table shows. Every entry is a restatement of a result of Real Algebra, Complex Algebra, Split-Complex Algebra, Dual-Numbers Algebra, Quaternion Algebra, Split-Quaternion Algebra, Biquaternion Algebra and Split-Biquaternion Algebra, cited to its section; nothing below is proved here.

The comparison is arranged so that the ladder of the eight algebras is climbed twice. The first climb is through the defining data — the presentation, the dimension, the centre, commutativity and associativity — where the algebras differ by the adjunction of a square root and its sign. The second climb is through the consequences of that data — the division property, the zero divisors, the idempotents, the ideals, the conjugations and the norm — where the sign of the adjunction is seen to decide the algebra, the topology of its units and its place in the ladder. The two biquaternion systems close the comparison, and their tensor-product description is what makes the last rung intelligible.

Defining Data

The following table compares the defining data of the eight algebras: the presentation, the real dimension, the centre and its real dimension, commutativity and associativity. The eight algebras are the columns, in the fixed order.

defining data	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
presentation	base field	$\mathbb{R}[i]$, $i^2=-1$	$\mathbb{R}[j]$, $j^2=+1$	$\mathbb{R}[\varepsilon]$, $\varepsilon^2=0$	$e_k^2=-1$, $e_ke_l=-e_le_k$	$e_1^2=-1$, $e_2^2=e_3^2=+1$	$\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	$\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$
real dimension	$1$	$2$	$2$	$2$	$4$	$4$	$8$	$8$
centre	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{R}$	$\mathbb{R}$	$\mathbb{C}_{\mathbb{B}}\cong\mathbb{C}$	$\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}\cong\mathbb{D}$
centre, real dimension	$1$	$2$	$2$	$2$	$1$	$1$	$2$	$2$
commutative	yes	yes	yes	yes	no	no	no	no
associative	yes	yes	yes	yes	yes	yes	yes	yes

The table shows the descent of the centre and the loss of commutativity as two independent rungs. The centre of $\mathbb{R}$ is $\mathbb{R}$ (Real Algebra, §Real Algebra); the centres of $\mathbb{C}$, $\mathbb{D}$ and $\mathbb{D}'$ are the algebras themselves, since each of the three is commutative (Complex Algebra, §Complex Algebra; Split-Complex Algebra, §The Split Complex Algebra; Dual-Numbers Algebra, §The Dual Numbers). At the quaternion rung the centre collapses to the real line — $Z(\mathbb{H}) = \mathbb{R}$ (Quaternion Algebra, §The Conjugate Decomposition), and $Z(\mathbb{H}_{\mathrm{s}}) = \mathbb{R}\cdot 1$, the scalar line $S$ (Split-Quaternion Algebra, §The Centre and Simplicity) — even though the dimension has doubled. The last rung restores a two-dimensional centre, but a different one: $Z(\mathbb{B}) = \mathbb{C}e_0 \cong \mathbb{C}$, the central imaginary line of the complexification (Biquaternion Algebra, §The Algebra Structure), and $Z(\mathbb{H}_{\mathbb{D}}) = \mathbb{D}_{\mathbb{H}_{\mathbb{D}}} \cong \mathbb{D}$, the idempotent span of $1$ and $j$ (Split-Biquaternion Algebra, §The Split Complex Subspace). Commutativity is lost once, at the quaternion rung, and never returns; associativity holds in all eight columns and is the property the octonions would be the first to forfeit, which is why the ladder stops where it does.

Algebraic Properties

The following table compares the properties that the defining data determine: the division property, the zero divisors, the idempotents and the Peirce decomposition they induce, the ideals and simplicity, the conjugations, the number of distinguished fixed-point subspaces, and the norm with its definiteness. The eight algebras are the columns, in the fixed order.

property	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
division algebra	yes	yes	no	no	yes	no	no	no
zero divisors	none	none	null lines $N=0$	nilpotent line $(\varepsilon)$	none	null cone $N=0$	zero-divisor cone $N=0$	one idempotent component zero
idempotents	$0,1$	$0,1$	$\tilde\Pi_\pm$	$0,1$	$0,1$	$\tilde\pi_\pm$	$\tilde\Pi_\xi$, $\xi^2=-1$	$\tilde\Pi_\pm$
Peirce decomposition	trivial	trivial	$\mathbb{D}=\mathbb{D}\tilde\Pi_+\oplus\mathbb{D}\tilde\Pi_-$	none	trivial	into two minimal left ideals	$\mathbb{B}=I_1\oplus I_2$	$\mathbb{H}_{\mathbb{D}}=\mathbb{H}\tilde\Pi_+\oplus\mathbb{H}\tilde\Pi_-$
ideals, simplicity	field, simple	field, simple	not simple, semisimple	local, not simple, not semisimple	simple, division	simple, not division	simple, not division	not simple, semisimple
conjugations	$1$	$2$	$2$	$2$	$3$	$3$	$4$	$4$
fixed-point subspaces	$1$	$2$	$2$	$2$	$2$	$4$	$6$	$4$
norm	$a^2$	$a^2+b^2$	$a^2-b^2$	$a^2$	$\sum q_k^2$	$q_0^2+q_1^2-q_2^2-q_3^2$	$\sum Q_\mu^2$	$\sum Q_\mu^2$
definiteness	positive definite	positive definite	indefinite $(1,1)$	degenerate, rank $1$	positive definite $(4,0)$	indefinite $(2,2)$	complex-valued	split complex-valued, anisotropic

The division column is the first rung of the second climb, and it follows the norm exactly. The three algebras whose norm is definite are the three division algebras — $\mathbb{R}$ with $N(a)=a^2$ (Real Norm and Invertibility, §The Norm), $\mathbb{C}$ with $N(Z)=a^2+b^2$ (Complex Norm and Invertibility, §The Norm) and $\mathbb{H}$ with $N(\tilde q)=\sum q_k^2$ (Quaternion Norm and Invertibility, §The Norm) — and the passage from $\mathbb{C}$ to $\mathbb{D}$ is the exchange of one sign in the norm, $a^2+b^2$ becoming $a^2-b^2$, which makes the form isotropic and the algebra no longer a division algebra (Split-Complex Norm and Invertibility, §The Norm). The dual numbers carry this degeneration to its limit: the norm $a^2$ is the square of a single coordinate, its vanishing set is the whole nilpotent line $\mathbb{R}\varepsilon$, and the algebra is local with maximal ideal $\mathbb{R}\varepsilon$ (Dual-Numbers Norm and Invertibility, §The Norm). The split quaternions are the four-dimensional case of the same phenomenon: the norm $q_0^2+q_1^2-q_2^2-q_3^2$ is indefinite of signature $(2,2)$, and the algebra is not a division algebra, with a null cone in place of the origin of $\mathbb{H}$ (Split-Quaternion Norm and Invertibility, §The Norm).

The idempotent column explains why the last rung splits. $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$ have only the trivial idempotents $0$ and $1$, which is the idempotent form of being a field or a division algebra; $\mathbb{D}$ has the pair $\tilde\Pi_\pm = \tfrac12(1\pm j)$ with $\tilde\Pi_+\tilde\Pi_-=0$ and $\tilde\Pi_++\tilde\Pi_-=1$, giving the ring isomorphism $\mathbb{D}\cong\mathbb{R}\oplus\mathbb{R}$ and the Peirce decomposition of Split-Complex Algebra, §The Idempotent Decomposition; $\mathbb{D}'$ has none besides $0$ and $1$ because it is local; $\mathbb{H}_{\mathrm{s}}$ has the non-central idempotents $\tilde\pi_\pm=\tfrac12(1\pm e_2)$, which cut the algebra into two minimal left ideals (Split-Quaternion Algebra, §The Idempotents and the Split-Complex Subspaces); $\mathbb{B}$ has the idempotents $\tilde\Pi_\xi=\tfrac12(e_0+\xi i)$ indexed by the roots $\xi$ of $-1$, and a primitive orthogonal pair among them cuts the algebra into the two minimal left ideals $I_1,I_2$ (Biquaternion Ideals and Peirce Decomposition, §Idempotents and Orthogonal Idempotents); and $\mathbb{H}_{\mathbb{D}}$ has the central idempotents $\tilde\Pi_\pm=\tfrac12(1\pm j)$, which split it as $\mathbb{H}\tilde\Pi_+\oplus\mathbb{H}\tilde\Pi_-\cong\mathbb{H}\oplus\mathbb{H}$ (Split-Biquaternion Algebra, §The Algebra Structure). The distinction between the split-complex idempotents of $\mathbb{D}$ and the split-biquaternion idempotents of $\mathbb{H}_{\mathbb{D}}$ is that the first are central in a commutative algebra and the second are central in a non-commutative one: the split at the last rung is a split into two ideals, whereas the split at the $\mathbb{D}$ rung is the resolution of an algebra into two fields.

The simplicity column divides the eight into three groups. The fields $\mathbb{R}$ and $\mathbb{C}$, the division algebra $\mathbb{H}$, and the two split systems $\mathbb{H}_{\mathrm{s}}$ and $\mathbb{B}$ are simple, the last two differing from the three division algebras only in having zero divisors (Split-Quaternion Algebra, §The Centre and Simplicity; Biquaternion Ideals and Peirce Decomposition, §The Two-Sided Ideals: Simplicity of $\mathbb{B}$). The split complex algebra is not simple but is semisimple: the idempotents $\tilde\Pi_\pm$ give $\mathbb{D}\cong\mathbb{R}\oplus\mathbb{R}$, a product of two fields with zero Jacobson radical (Split-Complex Algebra, §The Idempotent Decomposition). The dual numbers are neither simple nor semisimple: they are local with maximal ideal $\mathrm{M}=(\varepsilon)$, and every element of $\mathrm{M}$ is nilpotent of index two, so the Jacobson radical is nonzero (Dual-Numbers Algebra, §The Maximal Ideal). The split biquaternions are not simple but are semisimple, splitting as $\mathbb{H}\tilde\Pi_+\oplus\mathbb{H}\tilde\Pi_-\cong\mathbb{H}\oplus\mathbb{H}$ into two simple ideals (Split-Biquaternion Algebra, §Basic Properties). The pattern is that the tensor product with a commutative factor is simple exactly when that factor is a field, and that the dual numbers are the one case which fails even semisimplicity, because $\mathbb{D}'$ has no nontrivial idempotent to split along and its only proper ideal is nilpotent.

The conjugations column counts the distinguished involutions of each algebra, and the fixed-point-spaces column counts the real subspaces they cut out. $\mathbb{R}$ has only the identity and its one fixed space, the algebra itself, which is why the column of $\mathbb{R}$ is empty in the subspace tables of Comparison of Subspaces and Involutions. $\mathbb{C}$, $\mathbb{D}$ and $\mathbb{D}'$ have the single nontrivial conjugation of a quadratic algebra and its two eigenspaces. The quaternion rung adds the negation maps $\tilde q\mapsto-\tilde{q}^{\natural}$ and $\tilde q\mapsto-\tilde q$, so that $\mathbb{H}$ carries three conjugations with the two distinguished subspaces $\mathbb{R}_{\mathbb{H}}$ and $\operatorname{Im}\mathbb{H}$ (Quaternion Algebra, §Quaternions; The Scalar and Vector Subspaces of $\mathbb{H}$, §Definition and Basis). $\mathbb{H}_{\mathrm{s}}$ carries the conjugation ${}^{\natural}$, the principal involution $\alpha$ and the reversal $\rho$, giving the four subspaces $S, V, \mathbb{D}_2, \mathbb{D}_3$ (Split-Quaternion Relations Between Subspaces, §The Four Subspaces at a Glance). $\mathbb{B}$ carries four conjugations and has the six distinguished subspaces $\mathbb{C}_{\mathbb{B}}, \operatorname{Vect}(\mathbb{B}), \mathbb{H}_{\mathbb{B}}, i\mathbb{H}_{\mathbb{B}}, \mathbb{M}_+, \mathbb{M}_-$ of Biquaternion Relations Between Subspaces, §The Six Subspaces at a Glance; $\mathbb{H}_{\mathbb{D}}$ carries four conjugations and has the four subspaces of Split-Biquaternion Relations Between Subspaces, §The Four Subspaces at a Glance. The quaternion rung is thus where the involution group first has more than one nontrivial element, and the eight-dimensional rung is where it is largest.

The Units and the Zero Divisors

The following table compares the group of units of the eight algebras: its description, its number of connected components, and its compactness. The units are the elements whose norm does not vanish, which is the invertibility criterion restated in Comparison of Norms and Invertibility.

units	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
unit group	$\mathbb{R}\setminus\{0\}$	$\mathbb{C}\setminus\{0\}$	$\{N\neq 0\}$	$\{a\neq 0\}$	$\mathbb{H}\setminus\{0\}$	$\{N\neq 0\}$	$\{N\neq 0\}$	$N$ a unit of $\mathbb{D}$
description	$\{\pm1\}\times\mathbb{R}_{>0}$	$S^1\times\mathbb{R}_{>0}$	$\{\pm1\}^2\times\mathbb{R}^2$	$\mathbb{R}^{\times}\times\mathbb{R}$	$\mathbb{R}_{>0}\times S^3$	$\cong GL_2(\mathbb{R})$	$\cong GL(2,\mathbb{C})$	$\cong\mathbb{H}^{\times}\times\mathbb{H}^{\times}$
components	$2$	$1$	$4$	$2$	$1$	$2$	$1$	$1$
compact	no	no	no	no	no	no	no	no
maximal compact	$\{\pm1\}$	$S^1$	$\{\pm1\}^2$	$\{\pm1\}$	$S^3\cong Sp(1)$	$\{\pm1\}$	$U(2)$	$S^3\times S^3$

The table is the algebra side of the polar decomposition of each system, and its organising thread is the number of components. $\mathbb{R}^{\times}$ and $\mathbb{C}^{\times}$ are the classical cases: the first is disconnected, its two components being the positive and the negative half-line, and the second is connected, its unit circle compact (Complex Exponential and Lie Group Structure, §The Group of Units). The split complex algebra is the first with four components, because each of the two idempotent coordinates has an independent sign, and the dual numbers have two, the two halves $a>0$ and $a<0$ of the real line, the infinitesimal direction being a nilpotent added to each (Split-Complex Norm and Invertibility, §The Group of Units; Dual-Numbers Norm and Invertibility, §The Group of Units). The quaternion units are connected, with maximal compact subgroup the unit sphere $S^3\cong Sp(1)$ (Quaternion Exponential and Lie Group Structure, §The Group of Units); the split quaternions are disconnected with two components and maximal compact subgroup $\{\pm1\}$, the identity component being $\mathrm{SL}_2(\mathbb{R})$ (Split-Quaternion Exponential and Lie Group Structure, §The Group of Units). The two biquaternion systems are again connected but no longer compact: $\mathbb{B}^{\times}\cong GL(2,\mathbb{C})$, with maximal compact subgroup $U(2)$, and $\mathbb{H}_{\mathbb{D}}^{\times}\cong\mathbb{H}^{\times}\times\mathbb{H}^{\times}$, with maximal compact subgroup $S^3\times S^3$ (The Biquaternion Unit Group as a Topological Group, §The Group of Units; Split-Biquaternion Exponential and Lie Group Structure, §The Group of Units). No group of units in the table is compact, because each contains the scalars $\mathbb{R}_{>0}$; the compact groups of the last row are the maximal compact subgroups inside them.

The zero divisors are the nonzero non-invertible elements, and their geometry is the geometry of the vanishing set of the norm where the norm is scalar-valued, and of the split complex structure where it is not. In $\mathbb{D}$ they are the two null lines $\mathbb{R}(1\pm j)$, in $\mathbb{D}'$ the nilpotent line $\mathbb{R}\varepsilon$, in $\mathbb{H}_{\mathrm{s}}$ the null cone, and in $\mathbb{B}$ the cone $N=0$ of real codimension two, the single complex equation $Q_0^2+Q_1^2+Q_2^2+Q_3^2=0$ being two real equations. In $\mathbb{H}_{\mathbb{D}}$ the norm is anisotropic and its vanishing set is only the origin, so the zero divisors are not read off from it: they are the elements with exactly one idempotent component zero, equivalently those whose norm is a nonzero zero divisor of $\mathbb{D}$, and their union is the complement of the units, the two four-dimensional subspaces $\mathbb{H}\tilde\Pi_+$ and $\mathbb{H}\tilde\Pi_-$ (Split-Biquaternion Norm and Invertibility, §Invertibility). The shared feature is that the unit group is the complement of the zero-divisor set, and the set is the boundary at which the polar decomposition of the algebra fails.

The Two Biquaternion Systems as Tensor Products

The last rung of the ladder is described once here, because it is the reason the two eight-dimensional systems are the namesakes of the category. Both are tensor products of the quaternions with a two-dimensional commutative algebra over $\mathbb{R}$, and every contrast between them is the contrast between the two commutative factors.

datum	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
tensor description	—	—	—	—	—	—	$\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	$\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$
central unit	—	—	—	—	—	—	$i$, $i^2=-1$	$j$, $j^2=+1$
algebra structure	—	—	—	—	—	—	central simple over $\mathbb{C}$	$\cong\mathbb{H}\oplus\mathbb{H}$
splitting	—	—	—	—	—	—	$I_1\oplus I_2=V\oplus V$	$\mathbb{H}\tilde\Pi_+\oplus\mathbb{H}\tilde\Pi_-$

The six lower columns carry the marker — because none of them is a tensor product of $\mathbb{H}$ with a two-dimensional commutative algebra: the row is meaningful only for the two namesakes, and it is stated rather than omitted. For the two biquaternion systems the centre and the norm are recorded in the tables above and are not repeated here.

The two systems are built by adjoining to $\mathbb{H}$ the two distinct square roots of a central unit, $-1$ in $\mathbb{C}$ and $+1$ in $\mathbb{D}$, and the sign of the square root is the whole of the difference. The complexification is a field extension of the centre and produces a simple algebra (Biquaternion Algebra, §The Algebra Structure); the split case is a product of rings and produces a semisimple one, the product of two simple ideals each isomorphic to $\mathbb{H}$ (Split-Biquaternion Algebra, §Basic Properties). The two are the only columns of the master table whose norm is ring-valued and the only two whose analysis is built on a ring-valued Cauchy theory, and both properties follow from the single fact that $\mathbb{C}$ and $\mathbb{D}$ are the two quadratic algebras over $\mathbb{R}$ that admit a central imaginary or split-imaginary unit commuting with the quaternion basis.

Summary

The eight algebras agree in being associative, and differ in every other defining datum. Along the ladder the real dimension takes the values $1,2,2,2,4,4,8,8$; commutativity holds in the four commutative systems $\mathbb{R}, \mathbb{C}, \mathbb{D}, \mathbb{D}'$ and fails from the quaternion rung onward; the centre is the whole algebra in the four commutative cases, the scalar line in the two quaternion systems, and the two-dimensional central line $\mathbb{C}$ or $\mathbb{D}$ in the two biquaternion systems. The division property holds exactly where the norm is definite — $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$ — and fails in $\mathbb{D}, \mathbb{D}'$ and $\mathbb{H}_{\mathrm{s}}$ through isotropy or degeneracy and in $\mathbb{B}, \mathbb{H}_{\mathbb{D}}$ through a ring-valued norm, whose vanishing set on the zero divisors is a cone in $\mathbb{B}$ and the two ideals in $\mathbb{H}_{\mathbb{D}}$. Idempotents are trivial in the fields and the division algebra, the split-complex pair $\tilde\Pi_\pm$ in $\mathbb{D}$, the non-central pair $\tilde\pi_\pm$ in $\mathbb{H}_{\mathrm{s}}$, the family over the roots of $-1$ in $\mathbb{B}$, and the central pair $\tilde\Pi_\pm$ splitting $\mathbb{H}_{\mathbb{D}}$ into two copies of $\mathbb{H}$. Simplicity holds for $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{H}_{\mathrm{s}}, \mathbb{B}$, fails with semisimplicity for $\mathbb{D}$ and $\mathbb{H}_{\mathbb{D}}$, and fails without semisimplicity for $\mathbb{D}'$. The conjugations number $1,2,2,2,3,3,4,4$ and the distinguished fixed-point subspaces $1,2,2,2,2,4,6,4$. The two biquaternion systems are the tensor products $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ and $\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$, differing only in the sign of the central unit, and that sign decides whether the result is simple or a product.

Summary of Notation

symbol	meaning
$\mathbb{R}, \mathbb{C}, \mathbb{D}, \mathbb{D}'$	the four commutative systems, of dimensions $1,2,2,2$
$\mathbb{D}$	split complex numbers, $j^2=+1$
$\mathbb{D}'$	dual numbers, $\varepsilon^2=0$
$\mathbb{H}, \mathbb{H}_{\mathrm{s}}$	the two four-dimensional systems
$\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	biquaternions
$\mathbb{H}_{\mathbb{D}}=\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$	split biquaternions
$Z(A)$	the centre of an algebra $A$
$Z(\mathbb{H})=\mathbb{R}$, $Z(\mathbb{H}_{\mathrm{s}})=\mathbb{R}$	scalar centres of the quaternion systems
$\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_0$	the centre of $\mathbb{B}$
$\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$	the centre of $\mathbb{H}_{\mathbb{D}}$, $\cong\mathbb{D}$
$\tilde\Pi_\pm=\tfrac12(1\pm j)$	the idempotents of $\mathbb{D}$ and $\mathbb{H}_{\mathbb{D}}$
$\tilde\pi_\pm=\tfrac12(1\pm e_2)$	the non-central idempotents of $\mathbb{H}_{\mathrm{s}}$
$\tilde\Pi_\xi=\tfrac12(e_0+\xi i)$, $\xi^2=-1$	the idempotents of $\mathbb{B}$
$I_1,I_2$	the minimal left ideals of $\mathbb{B}$, $\mathbb{B}=I_1\oplus I_2$
$N$	the norm
$\mathbb{R}_{\mathbb{H}}, \operatorname{Im}\mathbb{H}$	the scalar and vector subspaces of $\mathbb{H}$
$S,V,\mathbb{D}_2,\mathbb{D}_3$	the four subspaces of $\mathbb{H}_{\mathrm{s}}$
$\mathbb{H}_{\mathbb{B}}, \mathbb{M}_+,\mathbb{M}_-$	subspaces of $\mathbb{B}$
`—`	the empty-cell marker of this category