Changes

\end{matrix}</math>

\end{matrix}</math>

This partition may also be expressed in the follwing symbolic form:

This partition may also be expressed in the following symbolic form:

: <math>\begin{matrix}

: <math>\begin{matrix}

X^2 & \cong & \operatorname{diag}(X) & + & 2 \tbinom{X}{2}.

X^2 & \cong & \operatorname{diag}(X) & + & 2 \tbinom{X}{2}.

\end{matrix}</math>

\end{matrix}</math>

We can now use the features in d<font face="lucida calligraphy">X</font> = {d''x''_''i''} = {d''x''₁,&nbsp;&hellip;,&nbsp;d''x''_''n''} to classify the paths of ('''B''' &rarr; ''X'') by way of the pairs in ''X''². If ''X'' <math>\cong</math> '''B'''^''n'' then a path in ''X'' has the form ''q'' : ('''B''' &rarr; '''B'''^''n'') <math>\cong</math> '''B'''^''n'' &times; '''B'''^''n'' <math>\cong</math> '''B'''^2''n'' <math>\cong</math> ('''B'''²)^''n''.  Intuitively, we want to map this ('''B'''²)^''n'' onto ''D''^''n'' by mapping each component '''B'''² onto a copy of '''D'''.  But in our current situation "'''D'''" is just a name we give, or an accidental quality we attribute, to coefficient values in '''B''' when they are attached to features in d<font face="lucida calligraphy">X</font>.

We may now use the features in <math>\operatorname{d}\mathcal{X} = \{ \operatorname{d}x_i \} = \{ \operatorname{d}x_1, \ldots, \operatorname{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math>  If ''X'' <math>\cong</math> '''B'''^''n'' then a path in ''X'' has the form ''q'' : ('''B''' &rarr; '''B'''^''n'') <math>\cong</math> '''B'''^''n'' &times; '''B'''^''n'' <math>\cong</math> '''B'''^2''n'' <math>\cong</math> ('''B'''²)^''n''.  Intuitively, we want to map this ('''B'''²)^''n'' onto ''D''^''n'' by mapping each component '''B'''² onto a copy of '''D'''.  But in our current situation "'''D'''" is just a name we give, or an accidental quality we attribute, to coefficient values in '''B''' when they are attached to features in d<font face="lucida calligraphy">X</font>.

Therefore, define d''x''_''i'' : ''X''² &rarr; '''B''' such that:

Therefore, define d''x''_''i'' : ''X''² &rarr; '''B''' such that: