Line 5,613: |
Line 5,613: |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%" | + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
− | |+ '''Table 55. Synopsis of Terminology : Restrictive and Alternative Subtypes''' | + | |+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math> |
− | |- style="background:ghostwhite" | + | |- style="height:40px; background:ghostwhite" |
− | !
| + | | |
− | ! Operator | + | | align="center" | <math>\text{Operator}\!</math> |
− | ! Proposition | + | | align="center" | <math>\text{Proposition}\!</math> |
− | ! Map | + | | align="center" | <math>\text{Map}\!</math> |
| |- | | |- |
| + | | align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\end{matrix}</math> |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Tacit
| + | \boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~ |
− | |-
| + | \boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\ |
− | | Extension
| + | \boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet) |
− | |}
| + | \end{array}</math> |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <math>\epsilon</math> :
| + | \boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\ |
− | |-
| + | \boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B} |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \end{array}</math> |
− | |-
| |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → ''X''<sup> •</sup>)
| |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | <math>\epsilon</math>''J'' :
| + | \boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\ |
| + | \boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1] |
| + | \end{array}</math> |
| |- | | |- |
− | | 〈''u'', ''v'', d''u'', d''v''〉 → '''B''' | + | | align="center" | <math>\begin{matrix}\underline\text{Trope}\\\text{extension}\end{matrix}</math> |
− | |- | |
− | | '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''B'''
| |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <math>\epsilon</math>''J'' :
| + | \eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~ |
− | |-
| + | \eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\ |
− | | [''u'', ''v'', d''u'', d''v''] → [''x'']
| + | \eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet) |
− | |-
| + | \end{array}</math> |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
| |
− | |}
| |
− | |-
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Trope
| + | \eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\ |
− | |-
| + | \eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D} |
− | | Extension
| + | \end{array}</math> |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <math>\eta</math> :
| + | \eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\ |
− | |-
| + | \eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1] |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \end{array}</math> |
| |- | | |- |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>) | + | | align="center" | <math>\begin{matrix}\underline\text{Enlargement}\\\text{operator}\end{matrix}</math> |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <math>\eta</math>''J'' :
| + | \mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~ |
− | |-
| + | \mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\ |
− | | 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
| + | \mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet) |
− | |-
| + | \end{array}</math> |
− | | '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
| |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <math>\eta</math>''J'' :
| + | \mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\ |
− | |-
| + | \mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D} |
− | | [''u'', ''v'', d''u'', d''v''] → [d''x'']
| + | \end{array}</math> |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
| |
− | |}
| |
− | |-
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Enlargement
| + | \mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\ |
| + | \mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1] |
| + | \end{array}</math> |
| |- | | |- |
− | | Operator | + | | align="center" | <math>\begin{matrix}\underline\text{Difference}\\\text{operator}\end{matrix}</math> |
− | |} | |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | E :
| + | \mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~ |
− | |-
| + | \mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\ |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet) |
− | |-
| + | \end{array}</math> |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
| |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | E''J'' :
| + | \mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\ |
− | |-
| + | \mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D} |
− | | 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
| + | \end{array}</math> |
− | |-
| |
− | | '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
| |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | E''J'' :
| + | \mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\ |
− | |-
| + | \mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1] |
− | | [''u'', ''v'', d''u'', d''v''] → [d''x'']
| + | \end{array}</math> |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
| |
− | |}
| |
− | |-
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | Difference
| |
− | |-
| |
− | | Operator
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | D :
| |
− | |-
| |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| |
− | |-
| |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | D''J'' :
| |
− | |-
| |
− | | 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
| |
− | |-
| |
− | | '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
− | | D''J'' :
| |
− | |-
| |
− | | [''u'', ''v'', d''u'', d''v''] → [d''x'']
| |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
| |
− | |}
| |
− | |-
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
− | | Differential
| |
− | |-
| |
− | | Operator
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | d :
| |
− | |-
| |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| |
− | |-
| |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
− | | d''J'' :
| |
− | |-
| |
− | | 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
| |
− | |-
| |
− | | '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
− | | d''J'' :
| |
− | |-
| |
− | | [''u'', ''v'', d''u'', d''v''] → [d''x'']
| |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
| |
− | |}
| |
| |- | | |- |
| + | | align="center" | <math>\begin{matrix}\underline\text{Differential}\\\text{operator}\end{matrix}</math> |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Remainder
| + | \mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~ |
− | |-
| + | \mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\ |
− | | Operator
| + | \mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet) |
− | |}
| + | \end{array}</math> |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | r :
| + | \mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\ |
− | |-
| + | \mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D} |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \end{array}</math> |
− | |-
| |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
| |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | r''J'' :
| + | \mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\ |
− | |-
| + | \mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1] |
− | | 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
| + | \end{array}\!</math> |
| |- | | |- |
− | | '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D''' | + | | align="center" | <math>\begin{matrix}\underline\text{Remainder}\\\text{operator}\end{matrix}\!</math> |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | r''J'' :
| + | \mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~ |
− | |-
| + | \mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\ |
− | | [''u'', ''v'', d''u'', d''v''] → [d''x'']
| + | \mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet) |
− | |-
| + | \end{array}</math> |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''D'''<sup>1</sup>]
| |
− | |}
| |
− | |-
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Radius
| + | \mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\ |
− | |-
| + | \mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D} |
− | | Operator
| + | \end{array}</math> |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
| + | \mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\ |
− | |-
| + | \mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1] |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \end{array}</math> |
| |- | | |- |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>) | + | | align="center" | <math>\begin{matrix}\underline\text{Radius}\\\text{operator}\end{matrix}</math> |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | |
| + | \mathsf{e} = (\boldsymbol\varepsilon, \eta) \\ |
− | |-
| + | \mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet) |
| + | \end{array}</math> |
| | | | | |
− | |-
| |
− | |
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
− | | <font face=georgia>'''e'''</font>''J'' :
| |
− | |-
| |
− | | [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
| |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
| |
− | |}
| |
− | |-
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Secant
| + | \mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\ |
| + | \mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1] |
| + | \end{array}</math> |
| |- | | |- |
− | | Operator | + | | align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math> |
− | |} | |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
| + | \mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\ |
− | |-
| + | \mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet) |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \end{array}</math> |
− | |-
| |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
| | | | | |
− | |-
| |
− | |
| |
− | |-
| |
− | |
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
− | | <font face=georgia>'''E'''</font>''J'' :
| |
− | |-
| |
− | | [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
| |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
| |
− | |}
| |
− | |-
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Chord
| + | \mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\ |
| + | \mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1] |
| + | \end{array}</math> |
| |- | | |- |
− | | Operator | + | | align="center" | <math>\begin{matrix}\underline\text{Chord}\\\text{operator}\end{matrix}</math> |
− | |} | |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
| + | \mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\ |
− | |-
| + | \mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet) |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \end{array}</math> |
− | |-
| |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| |
− | |
| |
− | |-
| |
− | |
| |
− | |-
| |
| | | | | |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | <font face=georgia>'''D'''</font>''J'' :
| + | \mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\ |
− | |-
| + | \mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1] |
− | | [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
| + | \end{array}</math> |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
| |
− | |}
| |
| |- | | |- |
| + | | align="center" | <math>\begin{matrix}\underline\text{Tangent}\\\text{functor}\end{matrix}</math> |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | Tangent
| + | \mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\ |
− | |-
| + | \mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet) |
− | | Functor
| + | \end{array}</math> |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
| + | \mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\ |
− | |-
| + | \mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D} |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | \end{array}</math> |
− | |-
| |
− | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
| |
− | |}
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{l} |
− | | d''J'' :
| + | \mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\ |
− | |-
| + | \mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1] |
− | | 〈''u'', ''v'', d''u'', d''v''〉 → '''D'''
| + | \end{array}</math> |
− | |-
| |
− | | '''B'''<sup>2</sup> × '''D'''<sup>2</sup> → '''D'''
| |
− | |}
| |
− | |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | <font face=georgia>'''T'''</font>''J'' :
| |
− | |-
| |
− | | [''u'', ''v'', d''u'', d''v''] → [''x'', d''x'']
| |
− | |-
| |
− | | ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B''' × '''D''']
| |
− | |}
| |
| |} | | |} |
| | | |