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nlab.ne
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nlab.ne
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# Core ==================================
_ -> " "
link[X] -> "<a href=\".\">" $X "</a>"
strong[X] -> "<strong>" $X "</strong>"
ml[X] -> $X | link[$X]
maybe[X] -> $X | null
maybe4[X] -> maybe[maybe2[$X]]
maybe8[X] -> maybe[maybe4[$X]]
either[X,Y] -> $X | $Y
scareQuote[X] -> "\"" $X "\""
# Math ==================================
mathfrak[X] -> "\\mathfrak{" $X "}"
mathcal[X] -> "\\mathcal{" $X "}"
mathscr[X] -> "\\mathscr{" $X "}"
mathbf[X] -> "\\mathbf{" $X "}"
overline[X] -> "\\overline{" $X "}"
widehat[X] -> "\\widehat{" $X "}"
hat[X] -> "\\hat{" $X "}"
dot[X] -> "\\dot{" $X "}"
ddot[X] -> "\\ddot{" $X "}"
decorate[X] -> overline[$X] | widehat[$X] | hat[$X] | dot[$X] | ddot[$X]
paren[X] -> "\\left(" $X "\\right)"
bracket[X] -> "\\left[" $X "\\right]"
group[X] -> paren[$X] | bracket[$X]
maybeGroup[X] -> $X | group[$X]
mLower -> "a" | "b" | "c" | "d" | "e" | "f" | "g" | "h" | "i" | "j" | "k" | "m" | "n" | "p" | "q" | "r" | "s" | "t" | "u" | "v" | "w" | "x" | "y" | "z" | "\\ell"
mUpper -> "A" | "B" | "C" | "D" | "E" | "F" | "G" | "H" | "I" | "J" | "K" | "L" | "M" | "N" | "O" | "P" | "Q" | "R" | "S" | "T" | "U" | "V" | "W" | "X" | "Y" | "Z"
mAnyChar -> mLower | mUpper
mAnyCharF -> mathfrak[mAnyChar] | mathcal[mAnyChar] | mathscr[mAnyChar] | mathbf[mAnyChar]
mAnyCharFA -> either[mAnyCharF, either[mAnyCharF, either[mAnyCharF, decorate[mAnyCharF]]]]
mBinOp -> "+" | "-" | "\\cdot" | "\\bullet" | "\\vee" | "\\wedge" | "\\backslash" | "\\circ"
mArrow -> "\\leftarrow" | "\\rightarrow" | "\\leftrightarrow" | "\\Leftarrow" | "\\Rightarrow" | "\\Leftrightarrow"
tex[X] -> "\\(" $X "\\)"
dtex[X] -> "\\[" $X "\\]"
mCategoryish -> mUpper | mAnyCharFA | mAnyCharFA
mFunctorish -> mCategoryish _ mArrow _ mCategoryish
mSmallExpr ->
"{" mSmallExpr "}_{" mSmallExpr "}"
| "{" mSmallExpr "}^{" mSmallExpr "}"
| mSmallExpr _ mBinOp _ mSmallExpr
| maybeGroup[mSmallExpr]
| mCategoryish
| mFunctorish
| mAnyCharFA
mBigOp -> "\\sum" | "\\prod" | "\\coprod" | "\\int" | "\\bigcap" | "\\bigcup" | "\\bigsqcup" | "\\bigvee" | "\\bigwedge" | "\\bigodot" | "\\bigotimes" | "\\bigoplus"
mBigOpFull -> mBigOp "_{" mLower maybe[":" _ mFunctorish] "}" maybe["^{" mAnyCharFA "}"]
mDisplayMath ->
mBigOpFull _ mSmallExpr
| mBigOpFull _ mSmallExpr _ mBinOp _ mBigOpFull _ mSmallExpr
| group[mBigOpFull _ mSmallExpr] _ "^{" mSmallExpr "}"
| mAnyChar _ group[mBigOpFull _ mSmallExpr]
mDisplayMathStatement ->
mDisplayMath _ "=" _ mDisplayMath
# Main ==================================
@include "noun-literals.ne"
@include "adjective-literals.ne"
description ->
ml[adverb] _ description
| ml[adj]
| tex[mSmallExpr] "-" adj
nounPhrase ->
maybe[description] _ ml[noun] _ maybe[tex[mSmallExpr]]
| "lift" _ maybe[tex[mSmallExpr]] _ "of a" _ maybe[description _] nounPhrase _ maybe[tex[mSmallExpr]]
| maybe[tex[mSmallExpr] either["-truncated", "-indexed"] _] nounPhrase maybe[tex[mSmallExpr]] _ "augmented with a" _ maybe[description _] nounPhrase
| maybe[tex[mSmallExpr] either["-truncated", "-indexed"] _] nounPhrase maybe[tex[mSmallExpr]] _ "together with a" _ maybe[description _] nounPhrase
nounPhrases ->
maybe[description] _ ml[nouns]
| ml[adj] _ ml[nouns]
| maybe[description] _ ml[nouns] _ "over the" _ ml[noun] _ "of" _ nounPhrases
| maybe[description] _ ml[nouns] _ "arising from" _ ml[adj] _ nounPhrases
connective -> "in other terms" | "generally" | "moreover" | "therefore" | "it follows that" | "similarly" | "sometimes" | "historically" | "note that" | "as such" | "trivially" | "in a sense" | "certainly" | "conversely" | "informally" | "usually"
openingConnective -> "generally" | "sometimes" | "historically" | "informally" | "in certain contexts"
iffy ->
"in the case that"
| "when"
| "for situations where"
| "provided that"
| "if"
inTheSense ->
"in the sense that"
| "in that"
| "corresponding to the fact that"
| "in the following manner: "
plainAlgebraicStructure -> "group" | "ring" | "field" | maybe["free" _] "module" | "vector space"
algebraicStructure -> plainAlgebraicStructure | plainAlgebraicStructure "oid"
vagueWord ->
"canonical"
| "standard"
| "structure-preserving"
| "visible"
| algebraicStructure "-like"
easyWord -> "obvious" | "trivial" | "evident" | "straightforward" | "definitional" | "elementary" | "appropriate" | "correctly chosen"
qualification ->
dtex[mDisplayMathStatement]
| "all" _ nounPhrases _ "commute"
| "the" _ easyWord _ "diagram commutes"
| "all the" _ easyWord _ "diagrams commute"
| nounPhrases _ "are" _ scareQuote[vagueWord] _ "from" _ tex[mUpper] "'s" _ "point of view"
| "the" _ easyWord _ nounPhrases _ "are" _ description
| tex[mSmallExpr] _ "is" _ description
| "the" _ description _ "object factors through" _ tex[mSmallExpr]
statement ->
"a" _ nounPhrase _ maybe["satisfying" _ dtex[mDisplayMathStatement]] "is always" _ description maybe[_ iffy _ qualification]
| "a" _ nounPhrase _ "satisfies the" _ description _ "property" maybe[_ iffy _ qualification]
| "a" _ nounPhrase _ "embeds" _ adverbPlain _ "into all" _ description _ nounPhrases
| "a" _ nounPhrase _ "is" _ description _ iffy _ qualification
| "all" _ nounPhrases _ "are" _ description maybe[_ inTheSense _ dtex[mDisplayMathStatement]]
| nounPhrases _ "are" _ nounPhrases
parentheticalBody ->
"where by" _ "\"" description "\"" _ "we mean" maybe[", of course,"] _ statement
| "meaning" _ tex[mSmallExpr] _ "is" _ description
parenthetical -> " (" parentheticalBody ") "
computed -> "computed" | "calculated" | "derived" | "written" | "made explicit"
sentenceBody ->
iffy _ qualification "," _ statement "."
| iffy _ qualification "," _ "it is said that" _ statement "."
| iffy _ tex[mSmallExpr] _ "is" _ description maybe[parenthetical] "," _ "then so is" _ tex[mSmallExpr] "."
| nounPhrases _ "may be" _ computed _ "using" _ nounPhrases _ "or" _ nounPhrases either[".", _ "by observing that" dtex[mDisplayMathStatement]]
| nounPhrases _ "may be" _ computed _ "using" _ nounPhrases _ iffy _ qualification "."
| "provided" _ qualification maybe[parenthetical] "," _ statement "."
| "the notion of a" _ nounPhrase _ "is an approximate solution to the problem of finding" _ nounPhrases _ "that satisfy" _ dtex[mDisplayMathStatement] maybe["with respect to" _ ml[nounPhrases]] "."
| "an analogous definition makes sense in the context of" _ nounPhrases maybe[parenthetical] "."
| "in" _ vagueWord _ nounPhrases maybe[parenthetical] "," _ statement "."
sentence -> maybe[connective "," _] sentenceBody _
openingSentence -> maybe[openingConnective "," _] sentenceBody _
defnBody ->
"A" _ strong[nounPhrase] _ "is a" _ nounPhrase _ "along with a" _ nounPhrase _ "that satisfies certain properties" either[".", ":" _ dtex[mDisplayMathStatement]]
| "A" _ strong[nounPhrase] _ "is a generalization of the notion of a" _ nounPhrase _ "into the context of" _ nounPhrases "."
| "In the context of" _ nounPhrases "," _ nounPhrases _ "are simply" _ nounPhrases _ "over" _ nounPhrases "."
| "A" _ nounPhrase _ "is called" _ strong[adj] _ iffy _ qualification "."
| nounPhrases _ "are said to be" _ strong[adj] _ iffy _ qualification "."
| "Given" _ tex[mSmallExpr] _ "and" _ tex[mSmallExpr] _ "," _ "a" _ nounPhrase _ "is" _ strong[adj] _ iffy _ qualification "."
defn -> strong["Definition"] ": " defnBody
title -> maybe[adverb] _ adj _ noun