Changes

Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as follows:

Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\operatorname{E}\mathcal{A}</math> as follows:

: <math>\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A} = \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n\}</math>

: <p><math>\begin{array}{lclcl}

\operatorname{E}\mathcal{A}

& = & \mathcal{A} \cup \operatorname{d}\mathcal{A}

& = & \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n\}. \\

\end{array}</math></p>

This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:

This supplies enough material to construct the ''differential extension'' <math>\operatorname{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:

:{| cellpadding=2

: <p><math>\begin{array}{lcl}

| E''A''

\operatorname{E}A

| =

& = & \langle \operatorname{E}\mathcal{A} \rangle \\

| ''A'' &times; d''A''

& = & \langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle  \\

& = & \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle, \\

| &nbsp;

\end{array}</math></p>

| 〈E<font face="lucida calligraphy">A</font>〉

| &nbsp;

| 〈<font face="lucida calligraphy">A</font> &cup; d<font face="lucida calligraphy">A</font>〉

| &nbsp;

| =

| 〈''a''₁,&nbsp;&hellip;,&nbsp;''a''_''n'',&nbsp;d''a''₁,&nbsp;&hellip;,&nbsp;d''a''_''n''〉,

thus giving E''A'' the type '''B'''^''n'' &times; '''D'''^''n''.

 

 

This gives <math>\operatorname{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>

Finally, the tangent universe E''A''^&nbsp;&bull; = [E<font face="lucida calligraphy">A</font>] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features E<font face="lucida calligraphy">A</font>:

Finally, the tangent universe E''A''^&nbsp;&bull; = [E<font face="lucida calligraphy">A</font>] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features E<font face="lucida calligraphy">A</font>: