diff --git a/tex/diagrams.tex b/tex/diagrams.tex index a9c8a0d..fd5f366 100644 --- a/tex/diagrams.tex +++ b/tex/diagrams.tex @@ -9,6 +9,7 @@ \usetikzlibrary{decorations.pathreplacing} \usetikzlibrary{decorations.pathmorphing} \usetikzlibrary{decorations.markings} +\usetikzlibrary{nfold} \tikzset{ desc/.style={sloped, fill=white,inner sep=2pt}, diff --git a/trees/tt-000X.tree b/trees/tt-000X.tree index 788402e..7e8b1eb 100644 --- a/trees/tt-000X.tree +++ b/trees/tt-000X.tree @@ -45,7 +45,7 @@ \li{coequalizer: \tikz{ \begin{tikzcd} - & \bullet \\ + & \anyobj \\ \\ & E \\ \bullet && \bullet @@ -62,7 +62,7 @@ \li{pushout (fiber coproduct): \tikz{ \begin{tikzcd} - \bullet \\ + \anyobj \\ \\ && {X +_{O} Y} && Y \\ \\ diff --git a/trees/tt-0013.tree b/trees/tt-0013.tree index 7e2ec9a..aa95a10 100644 --- a/trees/tt-0013.tree +++ b/trees/tt-0013.tree @@ -13,6 +13,8 @@ \transclude{tt-0015} +\transclude{tt-0041} + \transclude{tt-001D} % \transclude{tt-0016} diff --git a/trees/tt-001O.tree b/trees/tt-001O.tree index 84035bb..b5def51 100644 --- a/trees/tt-001O.tree +++ b/trees/tt-001O.tree @@ -12,6 +12,8 @@ A \vocab{set-valued} functor #{\fF: \C \to \Set} is called \newvocab{covariantly representable} if for some #{X \in \C}, ##{\tau: \fF \iso \C(X,-)} + +where #{\iso} denotes a \vocabk{natural isomorphism}{tt-001H}. } \p{Conversely, a set-valued functor #{\fG : \C^{op} \to \Set} is called \newvocab{contravariantly representable} if for some #{X \in \C}, diff --git a/trees/tt-003Y.tree b/trees/tt-003Y.tree index 9e153cf..9eeb944 100644 --- a/trees/tt-003Y.tree +++ b/trees/tt-003Y.tree @@ -12,13 +12,47 @@ \taxon{lemma}\refnotet{limit via representation}{6.1.1}{leinster2016basic}{ \p{ -Let #{\J} be a small category, #{\C} a category, and #{\fD: \J \to \C} a diagram. Then there is a one-to-one correspondence between limit cones on #{\fD} and representations of the \vocabk{cone functor}{tt-0040} +Let #{\J} be a small category, #{\C} a category, and #{\fD: \J \to \C} a diagram. Then there is a one-to-one correspondence between -##{\Cone(-, \fD): \C^{op} \to \Set} +\ul{ + \li{limit cones on #{\fD}} + \li{representations of the natural transformation \vocabk{Cone}{tt-0040}} +} + +with the representing objects being the limit objects (i.e. the vertices) of #{\fD}. -with the representing objects of #{\Cone(-, \fD)} being the limit objects (that is, the vertices of the limit cones) of #{\fD}. } -\p{Briefly put: a \vocabk{limit}{tt-002A} of #{\fD} is a \vocabk{representation}{tt-001O} of #{[\J, \C] (\Delta_{-}, \fD)}. +\p{Briefly put: a \vocabk{limit}{tt-002A} #{(V, \pi)} of #{\fD} is a \vocabk{representation}{tt-001O} of #{[\J, \C] (\Delta_{-}, \fD)}. +} + +\p{Schematically, +\tikz{ +\begin{tikzcd} + &&& \anyobj \\ + \\ + &&&& V \\ + J &&&&& \bullet \\ + \\ + {J'} &&&&& \bullet \\ + \J &&& \C + \arrow[dashed, from=1-4, to=3-5] + \arrow["{\pi_J}", from=3-5, to=4-6] + \arrow["{\pi_{J'}}"'{pos=0.2}, shorten >=6pt, from=3-5, to=6-6] + \arrow[""{name=0, anchor=center, inner sep=0}, from=4-1, to=6-1] + \arrow[""{name=1, anchor=center, inner sep=0}, from=4-6, to=6-6] + \arrow[""{name=2, anchor=center, inner sep=0}, "{\Delta_{\anyobj}}"{pos=0.8}, shorten <=12pt, Rightarrow, maps to, from=0, to=1-4] + \arrow["{\Delta_V}"{pos=0.8}, shorten <=13pt, Rightarrow, maps to, from=0, to=3-5] + \arrow[""{name=3, anchor=center, inner sep=0}, "\fD"', shorten <=17pt, Rightarrow, maps to, from=0, to=1] + \arrow["{[\J, \C] (\Delta_{-}, \fD)}"{pos=0.8}, shorten <=5pt, shorten >=5pt, Rightarrow, scaling nfold=3, maps to, from=2, to=3] +\end{tikzcd} +} +} + +\p{It implies that + +##{\Cone(\anyobj, \fD) \iso \C\left(\anyobj, \lim\limits_{\longleftarrow \J} \fD\right)} + +for any #{\anyobj \in \C}. } } \ No newline at end of file diff --git a/trees/tt-0040.tree b/trees/tt-0040.tree index 77660a2..94202a0 100644 --- a/trees/tt-0040.tree +++ b/trees/tt-0040.tree @@ -5,7 +5,7 @@ % definition theorem lemma construction observation % convention corollary axiom example exercise proof % discussion remark notation -\refdeft{cone functor}{eq. 6.1}{leinster2016basic}{ +\refdeft{cone as a natural transformation}{eq. 6.1}{leinster2016basic}{ \p{ Now, given a diagram #{\fD: \J \rightarrow \C} and an object #{V \in \C}, a \vocabk{cone}{tt-0028} on #{\fD} with vertex #{V} is simply a natural transformation from the \vocabk{diagonal functor}{tt-003T} #{\Delta_V} to the diagram #{\fD}. @@ -25,7 +25,7 @@ Now, given a diagram #{\fD: \J \rightarrow \C} and an object #{V \in \C}, a \voc \Cone(V, \fD)=[\J, \C] (\Delta_V, \fD) . }} -\p{Thus, #{\Cone(V, \fD)} is functorial in #{V} (contravariantly) and #{\fD} (covariantly). +\p{Thus, #{\Cone(V, \fD)} is \vocabk{functorial in}{tt-0041} #{V} (contravariantly) and #{\fD} (covariantly). } } diff --git a/trees/tt-0041.tree b/trees/tt-0041.tree new file mode 100644 index 0000000..2ea3c1a --- /dev/null +++ b/trees/tt-0041.tree @@ -0,0 +1,29 @@ +\import{tt-macros} +% clifford hopf spin tt math draft +\tag{tt} + +% definition theorem lemma construction observation +% convention corollary axiom example exercise proof +% discussion remark notation +\refdeft{functorial in}{sec. 4.1}{leinster2016basic}{ +\p{ +For some expression #{E(X)} containing #{X}, when we say #{E(X)} is (covariant) \newvocab{functorial in} #{X}, we mean that there exists a functor #{\fF} such that + +\tikz{ +\begin{tikzcd} + X &&& {E(X)} \\ + \\ + {X'} &&& {E(X')} + \arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=3-1] + \arrow[""{name=1, anchor=center, inner sep=0}, from=1-4, to=3-4] + \arrow["\fF", shorten <=20pt, shorten >=20pt, Rightarrow, maps to, from=0, to=1] +\end{tikzcd} +} + +for every #{f : X \to X'}. +} + +\p{Dually, we use the term \newvocab{contravariantly functorial in}. +} + +} \ No newline at end of file diff --git a/trees/tt-macros.tree b/trees/tt-macros.tree index 708611a..03d682f 100644 --- a/trees/tt-macros.tree +++ b/trees/tt-macros.tree @@ -55,7 +55,7 @@ \def\initobj{\mathrm{0}} \def\uniqobj{\mathrm{*}} -\def\anyobj{\mathrm{Z}} +\def\anyobj{\mathrm{-}} \def\obj{\operatorname{obj}} \def\Hom{\operatorname{Hom}}