From a4596781d37b336dca77777035cec3fc991b05b8 Mon Sep 17 00:00:00 2001 From: Samy Avrillon Date: Fri, 19 Jul 2024 20:35:26 +0200 Subject: [PATCH] Changed introduction --- Report/M2Report.tex | 204 ++++++---- Report/graphs/B1.json | 2 +- Report/graphs/D10.0.json | 2 +- Report/graphs/D10.json | 2 +- Report/graphs/D11.json | 2 +- Report/graphs/D4.json | 2 +- Report/graphs/D5.json | 2 +- Report/graphs/D9.json | 2 +- Report/graphs/G0.json | 2 +- Report/graphs/G1-0.json | 2 +- Report/graphs/G1.json | 2 +- Report/header.tex | 11 +- Report/yade.sty | 860 +++++++++++++++++++-------------------- 13 files changed, 573 insertions(+), 522 deletions(-) diff --git a/Report/M2Report.tex b/Report/M2Report.tex index 65efac2..8b0ffa3 100644 --- a/Report/M2Report.tex +++ b/Report/M2Report.tex @@ -23,10 +23,10 @@ \newpage - \section{Sort specification} + \section{Introduction} A Generalized Algebraic Theory (or GAT), first introduced by Cartmell \cite{CartmellGATs}, is a syntactic specification of an algebraic structure. From a GAT, one can define a category of models describing the models of the algebraic structure. - A GAT starts with a sort specification i.e. a list of sort declarations, eventually followed by a list of constructors. + A GAT typically starts with a sort specification i.e. a list of sort declarations, eventually followed by a list of constructors. In this document, we will not ask ourselves about the specific syntax of GATs, a \enquote{vague} definition is enough. \paragraph{Sort specification} @@ -46,9 +46,9 @@ A model of this category is a triple \begin{itemize} - \item A set $X_\Con : \Set$ - \item A family of sets $\left(X_\Ty\left(\Gamma\right)\right)_{\Gamma \in _\Con}$ - \item A family of sets $\left(X_\Tm\left(\Delta,A\right)\right)_{\Delta\in X_\Con,\: A \in X_\Ty\left(\Delta\right)}$ + \item A set $X_\Con$ of contexts + \item A family of sets $\left(X_\Ty\left(\Gamma\right)\right)_{\Gamma \in _\Con}$ of types, indexed by contexts + \item A family of sets $\left(X_\Tm\left(\Delta,A\right)\right)_{\Delta\in X_\Con,\: A \in X_\Ty\left(\Delta\right)}$ of terms, indexed by contexts and types. \end{itemize} \paragraph{Constructor specification} @@ -58,87 +58,62 @@ \vspace{1em} \renewcommand\arraystretch{1.5} \begin{tabular}{p{.37\textwidth}|p{.6\textwidth}} - $\operatorname{unit} : (\Gamma : \Con) \to \Ty\;\Gamma$ & In any context $\Gamma$, a type of $\Ty\;\Gamma$ called unit.\\ + $\operatorname{unit} : (\Gamma : \Con) \to \Ty\;\Gamma$ & In any context $\Gamma$, a type of $\Ty\;\Gamma$ called unit.\\ - $\operatorname{eq}: (\Gamma : \Con) \to (A : \Ty\;\Gamma) \to$ - - $\qquad\Tm\;\Gamma A \to \Tm\;\Gamma A \to \Ty\;\Gamma$ & In any context $\Gamma$ and type $A$ in this context, for every terms $t$,$u$ of the type $A$, we have a type $\operatorname{eq} \Gamma A t u$ describing the equality of the terms. + $\operatorname{tt}: (\Gamma : \Con) \to \Tm\;\Gamma\;(\operatorname{unit}\;\Gamma)$ & In any context $\Gamma$, we have a term whose type is the $\operatorname{unit}$ of this context ($\operatorname{unit}\;\Gamma$). \end{tabular} \vspace{1em} This adds to the previous model two functions that \enquote{points} one element of the sets \begin{itemize} \item For each $\Gamma \in X_\Con$, an element $\operatorname{unit}\;\Gamma \in X_\Ty(\Gamma)$ - \item For each $\Gamma \in X_\Con$, for each $A \in X_\Ty(\Gamma)$, for each elements $u,v \in X_\Tm(\Gamma,A)$, an element $\operatorname{eq}\;\Gamma\;A\;u\;v \in X_\Ty(\Gamma)$ + \item For each $\Gamma \in X_\Con$, an element $\operatorname{tt}\;\Gamma \in X_\Tm(\Gamma,\operatorname{unit}\;\Gamma)$ \end{itemize} Sort declarations describe the sets that the model contains, whereas the constructors describe elements of these sets. \paragraph{Two-sortification} - There is a process that allows us to transform a GAT into a GAT with only two sorts. This process is used by Philippo Sestini in his thesis \cite{SestiniPhD} refering the work of Zongpu Szumi Xie \cite{AmbrusSzumiXie2sort}: - - \begin{quote} - Many instances of multi-sorted IITs [IITs are another type of GATs] can be reduced to equivalent two-sorted IITs, via a systematic reduction method originally observed by Zongpu (Szumi) Xie. We are not aware of a formal proof of this construction for arbitrary IITs, but we conjecture that it does apply to all instances of induction-induction and consequently that it shows two-sorted IITs are enough to represent any specifiable IIT. - \end{quote} - - The goal of this document is to prove semantically that this transformation makes sense. More specifically, we prove that this transformation is a left adjunct functor of a coreflection. This is enough to prove what Sestini conjectured, i.e. that the initial object in the 2-sort category creates back the initial object of the primary category \cite[5. General]{nlab:reflective_subcategory}. - - We will now present this transformation. The sort specification of the transformed GAT is always the same, and contains two sort declarations (as planned): + There is a process that allows us to transform a GAT into a GAT with only two sorts. + The sort specification of the transformed GAT is always the same, and contains these two sort declarations: \vspace{1em} \begin{tabular}{p{0.37\textwidth}|p{0.5\textwidth}} - $\mathcal{O} : \Set$ & The set of sorts \\ - $\underline{\;\bullet\;} : \mathcal{O} \to \Set$ & For every sort object $o$ in the set of sorts, a set called $\underline{o}$ of objects corresponding to the sort object. + $\UU : \Set$ & The set of sorts \\ + $\El : \mathcal{O} \to \Set$ & For every sort object $o$ in the set of sorts, a set called $\El(o)$ of objects corresponding to the sort object. We will write this set $\underline{o}$ rather than $\El(o)$. \end{tabular} \vspace{1em} - Category of models of this two-sort specification are intuitively the category of families of set $\FamSet$, composed of pairs $\left(X_0:\Set,X_1: X_0 \to \Set\right)$. - - Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply underline to every parameter. For example, the Type Theory GAT presented above becomes that which follows: + Then, we replace all occurrences of $\Set$ to $\mathcal{O}$, and we apply $\El$ to every parameter. For example, the Type Theory GAT presented above becomes that which follows: \begin{tabular}{p{0.4\textwidth}|p{0.5\textwidth}} - $\Con : \mathcal{O}$ & One sort object is called \enquote{$\Con$} \\ + $\Con : \mathcal{O}$ & One sort object is called \enquote{$\Con$} \\ $\Ty : (\Gamma : \underline{\Con}) \to \mathcal{O}$ & For each object $\Gamma$ corresponding to the sort object $\Con$, another sort object called \enquote{$\Ty\;\Gamma$} \\ $\Tm : (\Gamma : \underline{\Con}) \to (A : \underline{\Ty\;\Delta}) \to \mathcal{O}$ & For each object $\Gamma$ corresponding to the sort object $\Con$, - and for every object $A$ corresponding to the sort object $\Ty\;\Gamma$, another sort object called \enquote{$\Tm\;\Gamma\;A$}\\ - $\operatorname{unit} : (\Gamma : \underline{\Con}) \to \underline{\Ty\;\Gamma}$ & - For each object $\Gamma$ corresponding to the sort object $\Con$, an object called \enquote{$\operatorname{unit} \Gamma$} corresponding to the sort object $\Ty\;\Gamma$\\ - $\operatorname{eq}: (\Gamma : \underline{\Con}) \to (A : \underline{\Ty\;\Gamma}) \to$ \newline - $\qquad\underline{\Tm\;\Gamma A} \to \underline{\Tm\;\Gamma A} \to \underline{\Ty\;\Gamma}$ & + and for every object $A$ corresponding to the sort object $\Ty\;\Gamma$, another sort object called "$\Tm\;\Gamma\;A$"\\ + $\operatorname{unit} : (\Gamma : \underline{\Con}) \to \underline{\Ty\;\Gamma}$ & + For each object $\Gamma$ corresponding to the sort object $\Con$, an object called "$\operatorname{unit} \Gamma$" corresponding to the sort object $\Ty\;\Gamma$\\ + $\operatorname{tt}: (\Gamma : \underline{\Con}) \to \underline{\Tm\;(\operatorname{unit}\;\Gamma)}$ & $\dots$ \end{tabular} - \paragraph{$\FamSet$ as functors} + This process has been observed by Zongpu Szumi Xie \cite{AmbrusSzumiXie2sort}, and Philippo Sestini used it in his thesis \cite{SestiniPhD}: - In the rest of the document, we will denote the simple category containing two elements and one non-identity arrow between them as $\TT$. The objects and arrow of this category are pictured below. - - \begin{center} - % YADE DIAGRAM G0.json - % GENERATED LATEX - \input{graphs/G0.tex} - % END OF GENERATED LATEX - \end{center} - - The functors over this categories are equivalent to families of sets, using the following mapping : - - \[ - \begin{array}{l|l} - X_\UU = X_0 & X_0 = X_\UU \\ - X_\El = \displaystyle\coprod_{A\in X_0}X_1(A) & X_1 = A \mapsto X_p^{-1}(\{A\})\\ - X_p = (A,B) \mapsto A & - \end{array} - \] - - Therefore the categories of sorts of the transformed GATs will be built atop of the category $\TSet$ rather than atop of the category $\FamSet$ as it makes the formal proofs more elegant. + \begin{quote} + Many instances of multi-sorted IITs [IITs are another type of GATs] can be reduced to equivalent two-sorted IITs, via a systematic reduction method originally observed by Zongpu (Szumi) Xie. We are not aware of a formal proof of this construction for arbitrary IITs, but we conjecture that it does apply to all instances of induction-induction and consequently that it shows two-sorted IITs are enough to represent any specifiable IIT. + \end{quote} \paragraph{Goal} - The goal of this document is to make a relation between the category of models of the GAT $\CC$ and the category of models of the two-sortified GAT $\BB$. This relation will be an adjunction $F \vdash G$ that we will prove to be a coreflection. + The goal my internship was to formally study this transformation, and to try to find a relation between the semantics of a GAT and its two-sorted version. - The category $\BB$ is built with an adjunction $R \vdash L$ to the category of models of the simple two-sort specification of sorts $\TSet$. + We managed to construct a coreflection between a category of models and the category of models of the transformed GAT. + The existence of this coreflection is enough to prove what Sestini conjectured. + + We will give a formal definition of this coreflection in next section. It will consist of an adjunction $F \vdash G$ between the category $\CC$ of the models of the GAT and the category $\BB$ of the models of the two-sortified GAT, where G is full and faithful. + The category $\BB$ will be built with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$. This forgetful $R$ functor is a composition of monadic functors, one for each sort constructor. \begin{center} % YADE DIAGRAM G1-0.json @@ -147,29 +122,104 @@ % END OF GENERATED LATEX \end{center} + \section{Construction of the coreflection} + + \subsection{Preliminaries} + + \paragraph{Category of models of the two-sort sort specification} + + The usual way of defining the category of models of the two-sort specification $\BB_0$ is by taking the category of families of sets. However, in order to have more elegant constructions, we will use a the category of models of the two-sort specification the category $\TSet$ of presheaves over the category with one arrow. + + In the rest of the document, we will denote this category with one arrow as $\TT$. The objects and arrow of this category are pictured below. + + \begin{center} + % YADE DIAGRAM G0.json + % GENERATED LATEX + \input{graphs/G0.tex} + % END OF GENERATED LATEX + \end{center} + + With this formalisation, a model of the two-sort GAT is a functor $X : \TSet$, such that + \begin{itemize} + \item $X_\UU$ is the set of the \enquote{sort objects} + \item For each sort object $\Gamma \in X_\UU$, the set of objects corresponding to the sort object is $X_p^-1(\{\Gamma\}) \subset X_\El$ + \end{itemize} + + Therefore the categories of models of the transformed GATs will be built atop of this category $\BB_0 = \TSet$. + + \paragraph{Grothendieck Construction} + For a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \Cat$, the Grothendieck construction is a category whose objects are pairs of + \begin{itemize} + \item $X$ an object of $\mathcal{C}$ + \item an object of $F(X)$ + \end{itemize} + + The morphism $(X,Y) \to (X',Y')$ is therefore a pair of a morphism $f : X \to X'$ in $\mathcal{C}$ and a morphism $g : F(f)(Y) \to Y'$ in $H(X')$. + + We will denote this category $(X : \mathcal{C}) \times F(X)$ as its objects are pairs. It can some times be found written as $\int^\mathcal{C} F$ or $\sqint^\mathcal{C} F$ + + \paragraph{Slice category} + For a category $\mathcal{C}$ and $X$ an object in that category, the slice category (or over category) $\mathcal{C}/A$ is a category whose objects are pairs of + \begin{itemize} + \item $Y$ an object of $\mathcal{C}$ + \item an arrow $X \to Y$ of $\mathcal{C}$ + \end{itemize} + + A morphism $(Y,f) \to (Y',f')$ is a morphism $g : Y \to Y'$ such that $g \circ f = f'$. + + We can deduce a functor $\left(\mathcal{C}/\dash\right) : \mathcal{C} \to \Cat$ from this construction. + + If the category $\mathcal{C}$ is $\Set$, we have that $\Set/X \cong \Set^X$. + + We will often concatenate the two method above to create from a category $\mathcal{C}$ and a functor $H : \mathcal{C} \to \Set$ a new category $(X : \mathcal{C}) \times \left(\Set\middle/H(X)\right)$. + + \paragraph{$\Hbar$ functor} + + Where $\Hbar_A$ is a functor $(X:C) \times (\Set/A(X)) \to \TSet$ defined as + \[ + \Hbar_X(X,(Y,f)) = \TSetObject{Y}{f}{A(X)} + \] + The morphisms are translated as-is. + + \begin{remark} + This functor can be constructed using the property of the Grothendieck construction + \end{remark} + \subsection{Constructing the categories} - We will construct both categories $\CC$ and $\BB$ recursively, adding new sorts one by one. - The categories $\CC_i$ are described as in Fiore's paper \cite{Fiore2008}, and the categories $\BB_i$ are constructed atop of the category $\TSet$ with a method inspired by the category of models described by Altenkirch et al. \cite{Altenkirch2018}. + We will construct both categories $\CC$ and $\BB$ recursively, adding new sorts one by one for each constructor. At each recursion step, we will build the categories, the adjunction, and keep some invariants that will be stated in \autoref{sec:hypotheses}. + + At the i-th recursion step, we will build the category $\CC_i$ which is the category of models of the $i$ first sorts of the sort specification. $\BB_i$ will samewise be the category of models of the 2-sorted $i$ first sorts of the sort specification. The overall recursive construction of the categories and of the adjunctions $F_i \vdash G_i$ is given below. \begin{center} % YADE DIAGRAM G1.json + % GENERATED LATEX + \input{graphs/G1.tex} + % END OF GENERATED LATEX \end{center} The first step of our recursion is the trivial adjunction $\lambda . \star \vdash \lambda . 1$ between the categories $\BB_0 = \TSet$ and $\CC_0 = 1$. + The functors $R_{i-1}^i$ are the forgetful monadic functors that forget about the $i$-th sort contsructor. They have a left adjoint denoted $L_{i-1}^i$. + As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will use the two following adjunctions + \[\begin{array}{c} + R_0^i = R_{0}^{i-1} \circ R_{i-1}^{i} = R_{0}^{1} \circ ... \circ R_{i-1}^{i}\\ + L_0^i = L_{i-1}^{i} \circ L_{0}^{i-1} = L_{i-1}^{i} \circ ... \circ L_{0}^{1} + \end{array}\] + + \begin{remark} + There is also an adjunction chain between $\CC_0$,$\dots$,$\CC_{i-1}$,$\CC_i$, but we don't use it in the proof. + \end{remark} + \subsubsection{Constructing $\CC_i$} We construct the category $\CC_i$ as the following pair: \[ \CC_i = (X : \CC_{i-1}) \times \left(\Set\middle/\Hom(O_i,X)\right) \] - where for any category $\mathcal{C}$ and $X$ an object of $\mathcal{C}$, $(\mathcal{C}/X)$ it the slice category (or over category) of arrows pointing out of $X$ (its objects $(Y,f)$ are arrows $f : X \to Y$ and its morphisms are morphisms creating commutative triangles).\inlinetodo{Assez clair ?} \inlinetodo{On ne voit pas que $(\Set/A(X)) \cong \Set^{A(X)}$} - and where $O_i$ is a specific object of the category $\CC_{i-1}$, such that $\Hom(O_i,X)$ is the set of parameters for the construction of the new sort. - \todo{Comment indiquer que la paire est dépendante ?} I will give now an example of those $O_i$ objects for our type theory example. We begin with \[ @@ -216,13 +266,7 @@ Identities and compositions are that of the category $\BB_{i-1}$, and categorical equalities are trivially derived from the diagram above. \paragraph{The adjunction} - We also define a functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$ that sends objects and morphisms to their first component. This functor is a \emph{right adjunct} of another functor we call $L_{i-1}^i$. - - As we can compose the adjunctions $R_0^1$,$R_1^2$,...,$R_{i-1}^i$, we will create the two following syntactic sugars for the composed adjunctions. - \[\begin{array}{c} - R_i^j = R_{i}^{i+1} \circ R_{i+1}^j = R_{i}^{j-1} \circ R_{j-1}^{j} = R_{i}^{i+1} \circ ... \circ R_{j-1}^{j}\\ - L_i^j = L_{j-1}^{j} \circ L_{i}^{j-1} = L_{i+1}^{j} \circ L_{i}^{i+1} = L_{j-1}^{j} \circ ... \circ L_{i}^{i+1} - \end{array}\] + We also define a functor $R_{i-1}^i : \BB_i \to \BB_{i-1}$ that sends objects and morphisms to their first component. This functor is a \emph{right adjoint} of another functor we call $L_{i-1}^i$. We will also denote $\eta_i^j : \mathbf{1} \to R_i^j L_i^j$ and $\varepsilon_i^j : L_i^j R_i^j \to \mathbf{1}$ to be the unit and counit of the adjunction $R_i^j \vdash L_i^j$. @@ -234,17 +278,19 @@ \end{remark} - \subsection{Some Hypotheses} + \subsection{Induction Hypotheses} In order to build and prove the adjunction, we will state some recurrence invariants that we will prove after having built objects. \begin{property}[H1] + + The canonical morphism \[ \simpleArrow{R_{i-1}^i X \oplus L_0^{i-1} Y}{\left\{R_{i-1}^i \inj_1^i ; R_{i-1}^i \inj_2^i \circ \eta_{i-1}^i\right\}}{R_{i-1}^i(X \oplus_i L_0^i Y)} \] - is an isomorphism. + that we will denote as $\en_{i-1}^i$ is an isomorphism. - Its recursive version is the following isomorphism + Its recursive version is the following isomorphism, denoted as $\en_0^i$ \[ \simpleArrow{ R_{0}^i X \oplus_0 Y}{\left\{R_0^i \inj_1^i ; R_0^i \inj_2^i \circ \eta_0^i\right\}}{R_0^i(X \oplus_i L_0^i Y)} \] @@ -291,16 +337,6 @@ W(X,Y) := \left(X \oplus L_0^{i-1} \Hbar_{\Hom(G_{i-1}O_i,\dash)}(X,Y), \widetilde{\inj_2} \right) \] - Where $\Hbar_A$ is a functor $(X:C) \times (\Set/A(X)) \to \TSet$ defined as - \[ - \Hbar_X(X,(Y,f)) = \TSetObject{Y}{f}{A(X)} - \] - The morphisms are translated as-is. - - \begin{remark} - This functor can be constructed as a lax colimit seeing elements of $A(X)/\Set$ as lax cocones over the diagram $\left[1 \xrightarrow{A(X)} \Set\right]$ in $\Cat$, and seeing elements of $\TSet$ as lax cocones over the diagram with no arrow $\left[\Set \quad \Set\right]$. \inlinetodo{Vérifier ça} - \end{remark} - With $\widetilde{\inj_2}$ being defined by \inlinetodo{Changer les noms des hypothèses H3' et H1r} \[ \begin{array}{lcl} @@ -351,7 +387,6 @@ % END OF GENERATED LATEX \end{center} - \subsection{Proof of the adjunction} We prove that $(W,E)$ make an adjunction showing that there is a natural isomorphism between $\Hom$ sets in both categories. @@ -456,7 +491,7 @@ Where $\varepsilon_i$ is the counit of the adjunction $R_{i-1}^i \vdash L_{i-1}^i$, going from $L_{i-1}^i R_{i-1}^i X$ to $X$. - This goes from $L_0^i Y = L_{i-1}^i L_0^{i-1} Y$ to $L_{i-1}^i(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y)$, which is equivalent to $L_{i-1}^i R_{i-1}^i X \oplus_i L_0^i Y)$ as $L_{i-1}^i$ is a left-adjunct functor and therefore it preserves colimits; then goes to $X \oplus_i L_0^i Y$. + This goes from $L_0^i Y = L_{i-1}^i L_0^{i-1} Y$ to $L_{i-1}^i(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y)$, which is equivalent to $L_{i-1}^i R_{i-1}^i X \oplus_i L_0^i Y)$ as $L_{i-1}^i$ is a left-adjoint functor and therefore it preserves colimits; then goes to $X \oplus_i L_0^i Y$. We will now show that this definition is actually a definition of the coproduct in $\BB_i$. To that extent, we take two objects $X$ and $Z$ in $\BB_i$, $Y$ in $\TSet$ and two morphisms of $\BB_i$ $\varphi_1 : X \to Z$ and $\varphi_2 : L_0^i Y \to Z$. @@ -754,6 +789,15 @@ + + + + + + + + + diff --git a/Report/graphs/B1.json b/Report/graphs/B1.json index 3d21311..c9f21d6 100644 --- a/Report/graphs/B1.json +++ b/Report/graphs/B1.json @@ -1 +1 @@ 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-\newunicodechar{λ}{{\lambda}} +\newfontface\russian{Liberation Serif} \newcommand\BB{{\ensuremath{\mathcal{B}}}} +\newcommand\en{{\text{\russian н}}} \newcommand\TT{{\ensuremath{\mathcal{T}}}} \newcommand\UU{{\ensuremath{\mathcal{U}}}} \newcommand\CC{{\ensuremath{\mathcal{C}}}} diff --git a/Report/yade.sty b/Report/yade.sty index 854e110..82c0de0 100644 --- a/Report/yade.sty +++ b/Report/yade.sty @@ -27,53 +27,53 @@ \usetikzlibrary{backgrounds} \usetikzlibrary{quotes} \pgfkeys{/pgf/.cd, - arity/.initial=4, + arity/.initial=4, } \def\twocell{% - \@ifnextchar[{\twocell@i}{\twocell@i[.4]}% + \@ifnextchar[{\twocell@i}{\twocell@i[.4]}% } \def\twocell@i[#1]{% - \@ifnextchar[{\twocell@ii[#1]}{\twocell@ii[#1][#1]}% + \@ifnextchar[{\twocell@ii[#1]}{\twocell@ii[#1][#1]}% } \def\twocell@ii[#1][#2]{% - \deuxcellule{#1}{#2}% + \deuxcellule{#1}{#2}% } % A TikZ style for curved arrows, inspired by AndréC's. \tikzset{curve/.style={settings={#1},to path={ - let \p1 = ($(\tikztostart) - (\tikztotarget)$) in - (\tikztostart) - .. controls - ($(\tikztostart)!\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*0.65pt}!270:(\tikztotarget)$) - and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*0.65pt}!270:(\tikztotarget)$) - .. (\tikztotarget)\tikztonodes}}, - settings/.code={\tikzset{yade/.cd,#1} - \def\pv##1{\pgfkeysvalueof{/tikz/yade/##1}}}, - yade/.cd,pos/.initial=0.35,ratio/.initial=0} + let \p1 = ($(\tikztostart) - (\tikztotarget)$) in + (\tikztostart) +.. controls +($(\tikztostart)!\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*0.65pt}!270:(\tikztotarget)$) +and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!{veclen(\x1,\y1)*\pv{ratio}*0.65pt}!270:(\tikztotarget)$) +.. (\tikztotarget)\tikztonodes}}, +settings/.code={\tikzset{yade/.cd,#1} +\def\pv##1{\pgfkeysvalueof{/tikz/yade/##1}}}, +yade/.cd,pos/.initial=0.35,ratio/.initial=0} \newcommand{\deuxcellule}[8]{% - \node[coordinate] (a) at (#3) {} ; % - \node[coordinate] (b) at (#4) {} ; % - \node[coordinate] (c) at (#5) {} ; % - \node[coordinate] (d) at (#6) {} ; % - \path let - \p1= ($(b) - (a)$) , % - \p2= ($(d) - (c)$), % - \n1={veclen(\x1,\y1)}, % - \n2={veclen(\x2,\y2)}, % - \p3=($(\x1/\n1,\y1/\n1)$),% - \p4=($(\x2/\n2,\y2/\n2)$),% - \n3={#1 * \n1},% - \p5=($(\n3 * \x3, \n3 * \y3)$),% - \n3={#2 * \n2},% - \p6=($(\n3 * \x4, \n3 * \y4)$) in% - (a) ++ (\p5) node[coordinate] (x) {} %% - (c) ++ (\p6) node[coordinate] (y) {} %% - ; - \draw[#7] (x) -| (y) ; % + \node[coordinate] (a) at (#3) {} ; % + \node[coordinate] (b) at (#4) {} ; % + \node[coordinate] (c) at (#5) {} ; % + \node[coordinate] (d) at (#6) {} ; % + \path let + \p1= ($(b) - (a)$) , % + \p2= ($(d) - (c)$), % + \n1={veclen(\x1,\y1)}, % + \n2={veclen(\x2,\y2)}, % + \p3=($(\x1/\n1,\y1/\n1)$),% + \p4=($(\x2/\n2,\y2/\n2)$),% + \n3={#1 * \n1},% + \p5=($(\n3 * \x3, \n3 * \y3)$),% + \n3={#2 * \n2},% + \p6=($(\n3 * \x4, \n3 * \y4)$) in% + (a) ++ (\p5) node[coordinate] (x) {} %% + (c) ++ (\p6) node[coordinate] (y) {} %% + ; + \draw[#7] (x) -| (y) ; % } % Old macros, for compatibility @@ -81,177 +81,177 @@ \newcommand{\twocellright}[5][.4]{\twocell[#1]{#2}{#3}{#4}{#5}{}{cell=0,bend right}} \DeclareDocumentCommand{\twocelll}{O{.4} o D(){0} > { \SplitArgument { 1 } { , } } D<>{0.5,0.5} m m o m m}{% - 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\deuxcellule{#1}{\IfNoValueTF{#2}{#1}{#2}}{#5}{#6}{\IfNoValueTF{#7}{#5}{#7}}{#8}{}{bend left={#3},celllr=#4,label={[above right]{$\scriptstyle #9$}}}} +\deuxcellule{#1}{\IfNoValueTF{#2}{#1}{#2}}{#5}{#6}{\IfNoValueTF{#7}{#5}{#7}}{#8}{}{bend left={#3},celllr=#4,label={[above right]{$\scriptstyle #9$}}}} \DeclareDocumentCommand{\twocellbl}{O{.4} o D(){30} > { \SplitArgument { 1 } { , } } D<>{0.5,0.5} m m o m m}{% - \deuxcellule{#1}{\IfNoValueTF{#2}{#1}{#2}}{#5}{#6}{\IfNoValueTF{#7}{#5}{#7}}{#8}{}{bend right={#3},celllr=#4,label={[below left]{$\scriptstyle #9$}}}} +\deuxcellule{#1}{\IfNoValueTF{#2}{#1}{#2}}{#5}{#6}{\IfNoValueTF{#7}{#5}{#7}}{#8}{}{bend right={#3},celllr=#4,label={[below left]{$\scriptstyle #9$}}}} % Two cell bent right \def\twocellrb{% - \@ifnextchar[{\twocellrb@i}{\twocellrb@i[.4]}% + \@ifnextchar[{\twocellrb@i}{\twocellrb@i[.4]}% } \def\twocellrb@i[#1]{% - \@ifnextchar[{\twocellrb@ii[#1]}{\twocellrb@ii[#1][#1]}% + \@ifnextchar[{\twocellrb@ii[#1]}{\twocellrb@ii[#1][#1]}% } \def\twocellrb@ii[#1][#2]{% - \deuxcellulerb{#1}{#2}% + \deuxcellulerb{#1}{#2}% } \newcommand{\deuxcellulerb}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labeld={#6}}} \def\twocellra{% - \@ifnextchar[{\twocellra@i}{\twocellra@i[.4]}% + \@ifnextchar[{\twocellra@i}{\twocellra@i[.4]}% } \def\twocellra@i[#1]{% - \@ifnextchar[{\twocellra@ii[#1]}{\twocellra@ii[#1][#1]}% + \@ifnextchar[{\twocellra@ii[#1]}{\twocellra@ii[#1][#1]}% } \def\twocellra@ii[#1][#2]{% - \deuxcellulera{#1}{#2}% + \deuxcellulera{#1}{#2}% } \newcommand{\deuxcellulera}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelu={#6}}} \def\twocellral{% - \@ifnextchar[{\twocellral@i}{\twocellral@i[.4]}% + \@ifnextchar[{\twocellral@i}{\twocellral@i[.4]}% } \def\twocellral@i[#1]{% - \@ifnextchar[{\twocellral@ii[#1]}{\twocellral@ii[#1][#1]}% + \@ifnextchar[{\twocellral@ii[#1]}{\twocellral@ii[#1][#1]}% } \def\twocellral@ii[#1][#2]{% - \deuxcelluleral{#1}{#2}% + \deuxcelluleral{#1}{#2}% } \newcommand{\deuxcelluleral}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelal={#6}}} \def\twocellro{% - \@ifnextchar[{\twocellro@i}{\twocellro@i[.4]}% + \@ifnextchar[{\twocellro@i}{\twocellro@i[.4]}% } \def\twocellro@i[#1]{% - \@ifnextchar[{\twocellro@ii[#1]}{\twocellro@ii[#1][#1]}% + \@ifnextchar[{\twocellro@ii[#1]}{\twocellro@ii[#1][#1]}% } \def\twocellro@ii[#1][#2]{% - \deuxcellulero{#1}{#2}% + \deuxcellulero{#1}{#2}% } \newcommand{\deuxcellulero}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelo={#6}}} \def\twocellrl{% - \@ifnextchar[{\twocellrl@i}{\twocellrl@i[.4]}% + \@ifnextchar[{\twocellrl@i}{\twocellrl@i[.4]}% } \def\twocellrl@i[#1]{% - \@ifnextchar[{\twocellrl@ii[#1]}{\twocellrl@ii[#1][#1]}% + \@ifnextchar[{\twocellrl@ii[#1]}{\twocellrl@ii[#1][#1]}% } \def\twocellrl@ii[#1][#2]{% - \deuxcellulerl{#1}{#2}% + \deuxcellulerl{#1}{#2}% } \newcommand{\deuxcellulerl}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labell={#6}}} \def\twocellrr{% - \@ifnextchar[{\twocellrr@i}{\twocellrr@i[.4]}% + \@ifnextchar[{\twocellrr@i}{\twocellrr@i[.4]}% } \def\twocellrr@i[#1]{% - \@ifnextchar[{\twocellrr@ii[#1]}{\twocellrr@ii[#1][#1]}% + \@ifnextchar[{\twocellrr@ii[#1]}{\twocellrr@ii[#1][#1]}% } \def\twocellrr@ii[#1][#2]{% - \deuxcellulerr{#1}{#2}% + \deuxcellulerr{#1}{#2}% } \newcommand{\deuxcellulerr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelr={#6}}} \def\twocellrbr{% - \@ifnextchar[{\twocellrbr@i}{\twocellrbr@i[.4]}% + \@ifnextchar[{\twocellrbr@i}{\twocellrbr@i[.4]}% } \def\twocellrbr@i[#1]{% - \@ifnextchar[{\twocellrbr@ii[#1]}{\twocellrbr@ii[#1][#1]}% + \@ifnextchar[{\twocellrbr@ii[#1]}{\twocellrbr@ii[#1][#1]}% } \def\twocellrbr@ii[#1][#2]{% - \deuxcellulerbr{#1}{#2}% + \deuxcellulerbr{#1}{#2}% } \newcommand{\deuxcellulerbr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend right,labelbr={#6}}} % Two cell bent left \def\twocelllb{% - \@ifnextchar[{\twocelllb@i}{\twocelllb@i[.4]}% + \@ifnextchar[{\twocelllb@i}{\twocelllb@i[.4]}% } \def\twocelllb@i[#1]{% - \@ifnextchar[{\twocelllb@ii[#1]}{\twocelllb@ii[#1][#1]}% + \@ifnextchar[{\twocelllb@ii[#1]}{\twocelllb@ii[#1][#1]}% } \def\twocelllb@ii[#1][#2]{% - \deuxcellulelb{#1}{#2}% + \deuxcellulelb{#1}{#2}% } \newcommand{\deuxcellulelb}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labeld={#6}}} \def\twocellla{% - \@ifnextchar[{\twocellla@i}{\twocellla@i[.4]}% + \@ifnextchar[{\twocellla@i}{\twocellla@i[.4]}% } \def\twocellla@i[#1]{% - \@ifnextchar[{\twocellla@ii[#1]}{\twocellla@ii[#1][#1]}% + \@ifnextchar[{\twocellla@ii[#1]}{\twocellla@ii[#1][#1]}% } \def\twocellla@ii[#1][#2]{% - \deuxcellulela{#1}{#2}% + \deuxcellulela{#1}{#2}% } \newcommand{\deuxcellulela}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelu={#6}}} \def\twocelllal{% - \@ifnextchar[{\twocelllal@i}{\twocelllal@i[.4]}% + \@ifnextchar[{\twocelllal@i}{\twocelllal@i[.4]}% } \def\twocelllal@i[#1]{% - \@ifnextchar[{\twocelllal@ii[#1]}{\twocelllal@ii[#1][#1]}% + \@ifnextchar[{\twocelllal@ii[#1]}{\twocelllal@ii[#1][#1]}% } \def\twocelllal@ii[#1][#2]{% - \deuxcellulelal{#1}{#2}% + \deuxcellulelal{#1}{#2}% } \newcommand{\deuxcellulelal}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelal={#6}}} \def\twocelllo{% - \@ifnextchar[{\twocelllo@i}{\twocelllo@i[.4]}% + \@ifnextchar[{\twocelllo@i}{\twocelllo@i[.4]}% } \def\twocelllo@i[#1]{% - \@ifnextchar[{\twocelllo@ii[#1]}{\twocelllo@ii[#1][#1]}% + \@ifnextchar[{\twocelllo@ii[#1]}{\twocelllo@ii[#1][#1]}% } \def\twocelllo@ii[#1][#2]{% - \deuxcellulelo{#1}{#2}% + \deuxcellulelo{#1}{#2}% } \newcommand{\deuxcellulelo}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelo={#6}}} \def\twocellll{% - \@ifnextchar[{\twocellll@i}{\twocellll@i[.4]}% + \@ifnextchar[{\twocellll@i}{\twocellll@i[.4]}% } \def\twocellll@i[#1]{% - \@ifnextchar[{\twocellll@ii[#1]}{\twocellll@ii[#1][#1]}% + \@ifnextchar[{\twocellll@ii[#1]}{\twocellll@ii[#1][#1]}% } \def\twocellll@ii[#1][#2]{% - \deuxcellulell{#1}{#2}% + \deuxcellulell{#1}{#2}% } \newcommand{\deuxcellulell}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labell={#6}}} \def\twocelllr{% - \@ifnextchar[{\twocelllr@i}{\twocelllr@i[.4]}% + \@ifnextchar[{\twocelllr@i}{\twocelllr@i[.4]}% } \def\twocelllr@i[#1]{% - \@ifnextchar[{\twocelllr@ii[#1]}{\twocelllr@ii[#1][#1]}% + \@ifnextchar[{\twocelllr@ii[#1]}{\twocelllr@ii[#1][#1]}% } \def\twocelllr@ii[#1][#2]{% - \deuxcellulelr{#1}{#2}% + \deuxcellulelr{#1}{#2}% } \newcommand{\deuxcellulelr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelr={#6}}} \def\twocelllbr{% - \@ifnextchar[{\twocelllbr@i}{\twocelllbr@i[.4]}% + \@ifnextchar[{\twocelllbr@i}{\twocelllbr@i[.4]}% } \def\twocelllbr@i[#1]{% - \@ifnextchar[{\twocelllbr@ii[#1]}{\twocelllbr@ii[#1][#1]}% + \@ifnextchar[{\twocelllbr@ii[#1]}{\twocelllbr@ii[#1][#1]}% } \def\twocelllbr@ii[#1][#2]{% - \deuxcellulelbr{#1}{#2}% + \deuxcellulelbr{#1}{#2}% } \newcommand{\deuxcellulelbr}[6]{\twocell[#1][#2]{#3}{#4}{#5}{}{cell=0,bend left,labelbr={#6}}} @@ -259,20 +259,20 @@ \newbox\xrat@below \newbox\xrat@above \newcommand{\Xarrow}[3][]{% - \setbox\xrat@below=\hbox{\ensuremath{\scriptstyle #2}}% - \setbox\xrat@above=\hbox{\ensuremath{\scriptstyle #3}}% - \pgfmathsetlengthmacro{\xrat@len}{max(\wd\xrat@below,\wd\xrat@above)+.6em}% - \mathrel{\tikz [#1,baseline=-.75ex] - \draw (0,0) -- node[below=-2pt] {\box\xrat@below} - node[above=-2pt] {\box\xrat@above} - (\xrat@len,0) ;}} + \setbox\xrat@below=\hbox{\ensuremath{\scriptstyle #2}}% + \setbox\xrat@above=\hbox{\ensuremath{\scriptstyle #3}}% + \pgfmathsetlengthmacro{\xrat@len}{max(\wd\xrat@below,\wd\xrat@above)+.6em}% + \mathrel{\tikz [#1,baseline=-.75ex] + \draw (0,0) -- node[below=-2pt] {\box\xrat@below} + node[above=-2pt] {\box\xrat@above} + (\xrat@len,0) ;}} \newcommand{\xarrow}[2][]{% - \setbox\xrat@above=\hbox{\ensuremath{\scriptstyle #2\ }}% - \pgfmathsetlengthmacro{\xrat@len}{\wd\xrat@above+.8em}% - \mathrel{\tikz [baseline=-.75ex] - \draw (0,0) edge[#1] node[above=-2pt] {\box\xrat@above} - (\xrat@len,0) ;}} + \setbox\xrat@above=\hbox{\ensuremath{\scriptstyle #2\ }}% + \pgfmathsetlengthmacro{\xrat@len}{\wd\xrat@above+.8em}% + \mathrel{\tikz [baseline=-.75ex] + \draw (0,0) edge[#1] node[above=-2pt] {\box\xrat@above} + (\xrat@len,0) ;}} @@ -281,7 +281,7 @@ \newenvironment{net}{\begin{tikzpicture}[baseline=(current bounding box.center),text depth=.2em,text height=.8em,inner sep=1pt]}{\end{tikzpicture}} \newcommand{\ssf}[2]{% - \inetatom(#1){#2}% +\inetatom(#1){#2}% } \newcommand{\gimpll}{\ssf{Rien}{\impll}} @@ -299,92 +299,92 @@ \pgfdeclaredecoration{single line}{initial}{ - \state{initial}[width=\pgfdecoratedpathlength-1sp]{\pgfpathmoveto{\pgfpointorigin}} - \state{final}{\pgfpathlineto{\pgfpointorigin}} + \state{initial}[width=\pgfdecoratedpathlength-1sp]{\pgfpathmoveto{\pgfpointorigin}} + \state{final}{\pgfpathlineto{\pgfpointorigin}} } \pgfdeclaredecoration{single line backwards}{initial}{ - \state{initial}[width=\pgfdecoratedpathlength-1sp]{\pgfpathmoveto{\pgfpointorigin}} - \state{final}{\pgfpathlineto{\pgfpointorigin}} + \state{initial}[width=\pgfdecoratedpathlength-1sp]{\pgfpathmoveto{\pgfpointorigin}} + \state{final}{\pgfpathlineto{\pgfpointorigin}} } \tikzset{ - raise line/.style={ - decoration={single line, raise=#1}, decorate - } + raise line/.style={ + decoration={single line, raise=#1}, decorate + } } \tikzset{ - mod/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with { - \path[draw,-] (0,-3pt) -- (0,3pt); +mod/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with { +\path[draw,-] (0,-3pt) -- (0,3pt); }}}}} \tikzset{ - label/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\node[inner sep=2pt,outer sep=0] #1 ;} +label/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with {\node[inner sep=2pt,outer sep=0] #1 ;} }}}} \tikzset{ - labelo/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\node[circle,inner sep=0pt,fill=white] {$\scriptstyle #1$} ;} +labelo/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with {\node[circle,inner sep=0pt,fill=white] {$\scriptstyle #1$} ;} }}}} \tikzset{ - labelon/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\node[inner sep=1pt,fill=white] {$\scriptstyle #1$} ;} +labelon/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with {\node[inner sep=1pt,fill=white] {$\scriptstyle #1$} ;} }}}} \tikzset{ - labelonb/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\node[inner sep=0pt,fill={bg}] {$\scriptstyle #1$} ;} +labelonb/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with {\node[inner sep=0pt,fill={bg}] {$\scriptstyle #1$} ;} }}}} \tikzset{ - labelat/.style 2 args={postaction={ - decorate, - decoration={ - markings, - mark=at position #2 with {\node[inner sep=2pt] #1 ;} +labelat/.style 2 args={postaction={ +decorate, +decoration={ +markings, +mark=at position #2 with {\node[inner sep=2pt] #1 ;} }}}} \tikzset{ - labeloat/.style 2 args={postaction={ - decorate, - decoration={ - markings, - mark=at position #2 with {\node[inner sep=0.1pt,fill=white] {$\scriptstyle #1$} ;} +labeloat/.style 2 args={postaction={ +decorate, +decoration={ +markings, +mark=at position #2 with {\node[inner sep=0.1pt,fill=white] {$\scriptstyle #1$} ;} }}}} \tikzset{ - labelonat/.style 2 args={postaction={ - decorate, - decoration={ - markings, - mark=at position #2 with {\node[inner sep=0.1pt,fill=white] {$\scriptstyle #1$} ;} +labelonat/.style 2 args={postaction={ +decorate, +decoration={ +markings, +mark=at position #2 with {\node[inner sep=0.1pt,fill=white] {$\scriptstyle #1$} ;} }}}} \tikzset{ - labelonatsloped/.style 2 args={postaction={ - decorate, - decoration={ - markings, - mark=at position #2 with {\node[inner sep=0.1pt,fill=white,transform shape] {#1} ;} +labelonatsloped/.style 2 args={postaction={ +decorate, +decoration={ +markings, +mark=at position #2 with {\node[inner sep=0.1pt,fill=white,transform shape] {#1} ;} }}}} @@ -393,22 +393,22 @@ \tikzset{diagnode/.style={anchor=base,inner sep=5pt,outer sep=0pt}} \tikzset{diag/.style 2 args=% - { - matrix of math nodes,ampersand replacement=\&, % - text height=1.2ex, text depth=0.25ex, % - row sep={#1 cm,between borders}, % - column sep={#2 cm,between borders}} + { +matrix of math nodes,ampersand replacement=\&, % +text height=1.2ex, text depth=0.25ex, % +row sep={#1 cm,between borders}, % +column sep={#2 cm,between borders}} }% \tikzset{diagorigins/.style 2 args=% - {matrix of math nodes,ampersand replacement=\&, % - row sep={#1 cm,between origins}, % - column sep={#2 cm,between origins}} +{matrix of math nodes,ampersand replacement=\&, % +row sep={#1 cm,between origins}, % +column sep={#2 cm,between origins}} }% \tikzset{stringdiag/.style 2 args=% - {nodes={inner sep=1pt,outer sep=0pt},% - ampersand replacement=\&,% - row sep={#1 cm,between origins}, % - column sep={#2 cm,between origins}% +{nodes={inner sep=1pt,outer sep=0pt},% + ampersand replacement=\&,% + row sep={#1 cm,between origins}, % + column sep={#2 cm,between origins}% }}% \newcommand{\diaggrandhauteur}{1} \newcommand{\diaggrandlargeur}{2} @@ -416,310 +416,310 @@ \newcommand{\diagpetitlargeur}{1.5} \tikzset{organigram/.style 2 args={matrix of nodes,ampersand replacement=\&, % - text height=1.7ex, text depth=0.25ex, % - row sep={#1 cm,between origins}, % - column sep={#2 cm,between origins} }} +text height=1.7ex, text depth=0.25ex, % +row sep={#1 cm,between origins}, % +column sep={#2 cm,between origins} }} \tikzset{graphe/.style 2 args={matrix of math nodes,ampersand replacement=\&, % - row sep={#1 cm,between origins}, % - column sep={#2 cm,between origins}, % - inner sep=-.1ex}} % + row sep={#1 cm,between origins}, % + column sep={#2 cm,between origins}, % + inner sep=-.1ex}} % \tikzset{ - two/.style 2 args={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with \node (#1) [#2] {} ; +two/.style 2 args={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with \node (#1) [#2] {} ; }}}} \tikzset{ - twocenter/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with \node (#1) {} ; +twocenter/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with \node (#1) {} ; }}}} \tikzset{ - twoon/.style 2 args={twocenter={#1},label={{$\scriptstyle #2$}}} +twoon/.style 2 args={twocenter={#1},label={{$\scriptstyle #2$}}} } \tikzset{ - twoo/.style={two={on},label={{$\scriptstyle #1$}}} +twoo/.style={two={on},label={{$\scriptstyle #1$}}} } \tikzset{ - twol/.style={two={l}{right},label={[left]{$\scriptstyle #1$}}} +twol/.style={two={l}{right},label={[left]{$\scriptstyle #1$}}} } \tikzset{ - twoleft/.style 2 args={two={#1}{right},label={[left]{$\scriptstyle #2$}}} +twoleft/.style 2 args={two={#1}{right},label={[left]{$\scriptstyle #2$}}} } \tikzset{ - twor/.style={two={r}{left},label={[right]{$\scriptstyle #1$}}} +twor/.style={two={r}{left},label={[right]{$\scriptstyle #1$}}} } \tikzset{ - tworight/.style 2 args={two={#1}{left},label={[right]{$\scriptstyle #2$}}} +tworight/.style 2 args={two={#1}{left},label={[right]{$\scriptstyle #2$}}} } \tikzset{ - twou/.style={two={u}{below},label={[above]{$\scriptstyle #1$}}} +twou/.style={two={u}{below},label={[above]{$\scriptstyle #1$}}} } \tikzset{ - twoa/.style={two={a}{below},label={[above]{$\scriptstyle #1$}}} +twoa/.style={two={a}{below},label={[above]{$\scriptstyle #1$}}} } \tikzset{ - twoup/.style 2 args={two={#1}{below},label={[above]{$\scriptstyle #2$}}} +twoup/.style 2 args={two={#1}{below},label={[above]{$\scriptstyle #2$}}} } \tikzset{ - twoabove/.style 2 args={two={#1}{below},label={[above]{$\scriptstyle #2$}}} +twoabove/.style 2 args={two={#1}{below},label={[above]{$\scriptstyle #2$}}} } \tikzset{ - twod/.style={two={d}{above},label={[below]{$\scriptstyle #1$}}} +twod/.style={two={d}{above},label={[below]{$\scriptstyle #1$}}} } \tikzset{ - twob/.style={two={b}{above},label={[below]{$\scriptstyle #1$}}} +twob/.style={two={b}{above},label={[below]{$\scriptstyle #1$}}} } \tikzset{ - twodown/.style 2 args={two={#1}{above},label={[below]{$\scriptstyle #2$}}} +twodown/.style 2 args={two={#1}{above},label={[below]{$\scriptstyle #2$}}} } \tikzset{ - twobelow/.style 2 args={two={#1}{above},label={[below]{$\scriptstyle #2$}}} +twobelow/.style 2 args={two={#1}{above},label={[below]{$\scriptstyle #2$}}} } \tikzset{ - twoal/.style={two={al}{right},label={[above left]{$\scriptstyle #1$}}} +twoal/.style={two={al}{right},label={[above left]{$\scriptstyle #1$}}} } \tikzset{ - twoaboveleft/.style 2 args={two={#1}{above},label={[above left]{$\scriptstyle #2$}}} +twoaboveleft/.style 2 args={two={#1}{above},label={[above left]{$\scriptstyle #2$}}} } \tikzset{ - twoar/.style={two={ar}{left},label={[above right]{$\scriptstyle #1$}}} +twoar/.style={two={ar}{left},label={[above right]{$\scriptstyle #1$}}} } \tikzset{ - twoaboveright/.style 2 args={two={#1}{above},label={[above right]{$\scriptstyle #2$}}} +twoaboveright/.style 2 args={two={#1}{above},label={[above right]{$\scriptstyle #2$}}} } \tikzset{ - twobr/.style={two={br}{left},label={[below right]{$\scriptstyle #1$}}} +twobr/.style={two={br}{left},label={[below right]{$\scriptstyle #1$}}} } \tikzset{ - twobelowright/.style 2 args={two={#1}{left},label={[below right]{$\scriptstyle #2$}}} +twobelowright/.style 2 args={two={#1}{left},label={[below right]{$\scriptstyle #2$}}} } \tikzset{ - twobl/.style={two={bl}{right},label={[below left]{$\scriptstyle #1$}}} +twobl/.style={two={bl}{right},label={[below left]{$\scriptstyle #1$}}} } \tikzset{ - twobelowleft/.style 2 args={two={#1}{right},label={[below left]{$\scriptstyle #2$}}} +twobelowleft/.style 2 args={two={#1}{right},label={[below left]{$\scriptstyle #2$}}} } \tikzset{ - twol/.style={two={l}{right},label={[left]{$\scriptstyle #1$}}} +twol/.style={two={l}{right},label={[left]{$\scriptstyle #1$}}} } \tikzset{ - labell/.style={label={[left]{$\scriptstyle #1$}}} +labell/.style={label={[left]{$\scriptstyle #1$}}} } \tikzset{ - labellat/.style 2 args={labelat={[left]{$\scriptstyle #1$}}{#2}} +labellat/.style 2 args={labelat={[left]{$\scriptstyle #1$}}{#2}} } \tikzset{ - labelr/.style={label={[right]{$\scriptstyle #1$}}} +labelr/.style={label={[right]{$\scriptstyle #1$}}} } \tikzset{ - labelrat/.style 2 args={labelat={[right]{$\scriptstyle #1$}}{#2}} +labelrat/.style 2 args={labelat={[right]{$\scriptstyle #1$}}{#2}} } \tikzset{ - labelar/.style={label={[above right]{$\scriptstyle #1$}}} +labelar/.style={label={[above right]{$\scriptstyle #1$}}} } \tikzset{ - labelarat/.style 2 args={labelat={[above right]{$\scriptstyle #1$}}{#2}} +labelarat/.style 2 args={labelat={[above right]{$\scriptstyle #1$}}{#2}} } \tikzset{ - labelbr/.style={label={[below right]{$\scriptstyle #1$}}} +labelbr/.style={label={[below right]{$\scriptstyle #1$}}} } \tikzset{ - labelbrat/.style 2 args={labelat={[below right]{$\scriptstyle #1$}}{#2}} +labelbrat/.style 2 args={labelat={[below right]{$\scriptstyle #1$}}{#2}} } \tikzset{ - labelu/.style={label={[above]{$\scriptstyle #1$}}} +labelu/.style={label={[above]{$\scriptstyle #1$}}} } \tikzset{ - labeluat/.style 2 args={labelat={[above]{$\scriptstyle #1$}}{#2}} +labeluat/.style 2 args={labelat={[above]{$\scriptstyle #1$}}{#2}} } \tikzset{ - labela/.style={label={[above]{$\scriptstyle #1$}}} + labela/.style={label={[above]{$\scriptstyle #1$}}} } \tikzset{ - labelaat/.style 2 args={labelat={[above]{$\scriptstyle #1$}}{#2}} +labelaat/.style 2 args={labelat={[above]{$\scriptstyle #1$}}{#2}} } \tikzset{ - loina/.style={label={[above=.5em]{$\scriptstyle #1$}}} + loina/.style={label={[above=.5em]{$\scriptstyle #1$}}} } \tikzset{ - labeld/.style={label={[below]{$\scriptstyle #1$}}} +labeld/.style={label={[below]{$\scriptstyle #1$}}} } \tikzset{ - labelb/.style={label={[below]{$\scriptstyle #1$}}} +labelb/.style={label={[below]{$\scriptstyle #1$}}} } \tikzset{ - labelbat/.style 2 args={labelat={[below]{$\scriptstyle #1$}}{#2}} +labelbat/.style 2 args={labelat={[below]{$\scriptstyle #1$}}{#2}} } \tikzset{ - loinb/.style={label={[below=.5em]{$\scriptstyle #1$}}} +loinb/.style={label={[below=.5em]{$\scriptstyle #1$}}} } \tikzset{ - labelal/.style={label={[above left]{$\scriptstyle #1$}}} +labelal/.style={label={[above left]{$\scriptstyle #1$}}} } \tikzset{ - labelalat/.style 2 args={labelat={[above left]{$\scriptstyle #1$}}{#2}} +labelalat/.style 2 args={labelat={[above left]{$\scriptstyle #1$}}{#2}} } \tikzset{ - labelbl/.style={label={[below left]{$\scriptstyle #1$}}} +labelbl/.style={label={[below left]{$\scriptstyle #1$}}} } \tikzset{ - labelblat/.style 2 args={labelat={[below left]{$\scriptstyle #1$}}{#2}} +labelblat/.style 2 args={labelat={[below left]{$\scriptstyle #1$}}{#2}} } \tikzset{ - labellat/.style 2 args={labelat={[left]{$\scriptstyle #1$}}{#2}} +labellat/.style 2 args={labelat={[left]{$\scriptstyle #1$}}{#2}} } \newcommand{\cs}[2][draw,->]{ % - \path[#1] (#2-1-1) -- (#2-1-2) ; % - \path[#1] (#2-1-3) -- (#2-1-2) ; % + \path[#1] (#2-1-1) -- (#2-1-2) ; % + \path[#1] (#2-1-3) -- (#2-1-2) ; % } \newcommand{\cospan}{\cs{m}} \newcommand{\vdoublecs}[2][draw,->]{ - \foreach \i in {1,2,3} % - {% - \path[#1] (#2-1-\i) -- (#2-2-\i) ;% - \path[#1] (#2-3-\i) -- (#2-2-\i) ; % - } ; % + \foreach \i in {1,2,3} % + {% + \path[#1] (#2-1-\i) -- (#2-2-\i) ;% + \path[#1] (#2-3-\i) -- (#2-2-\i) ; % + } ; % } \newcommand{\hdoublecs}[2][draw,->]{ - \foreach \i in {1,2,3} % - {% - \path[#1] (#2-\i-1) -- (#2-\i-2) ;% - \path[#1] (#2-\i-3) -- (#2-\i-2) ; % - } ; % + \foreach \i in {1,2,3} % + {% + \path[#1] (#2-\i-1) -- (#2-\i-2) ;% + \path[#1] (#2-\i-3) -- (#2-\i-2) ; % + } ; % } \newcommand{\vdoubleisos}[1]{ - \foreach \i in {1,2,3} % - {% - \isopath{#1-1-\i}{#1-2-\i} % - \isopath{#1-2-\i}{#1-3-\i} % - } ; % + \foreach \i in {1,2,3} % + {% + \isopath{#1-1-\i}{#1-2-\i} % + \isopath{#1-2-\i}{#1-3-\i} % + } ; % } \newcommand{\hdoubleisos}[1]{ - \foreach \i in {1,2,3} % - {% - \isopath{#1-\i-1}{#1-\i-2} % - \isopath{#1-\i-2}{#1-\i-3} % - } ; % + \foreach \i in {1,2,3} % + {% + \isopath{#1-\i-1}{#1-\i-2} % + \isopath{#1-\i-2}{#1-\i-3} % + } ; % } \newcommand{\doublecs}[2][draw,->]{% - \vdoublecs[#1]{#2} % - \hdoublecs[#1]{#2} % + \vdoublecs[#1]{#2} % + \hdoublecs[#1]{#2} % } \newcommand{\doublecospan}{\doublecs{m}} \newcommand{\sq}[4]{% - (m-1-1) edge[twou={#1}] (m-1-2) % - (m-1-1) edge[twol={#2}] (m-2-1) % - (m-1-2) edge[twor={#3}] (m-2-2) % - (m-2-1) edge[twod={#4}] (m-2-2) % + (m-1-1) edge[twou={#1}] (m-1-2) % + (m-1-1) edge[twol={#2}] (m-2-1) % + (m-1-2) edge[twor={#3}] (m-2-2) % + (m-2-1) edge[twod={#4}] (m-2-2) % } \newcommand{\sqpath}[6]{ - \draw[->,#6,rounded corners] - (#1) -- +(#2:#3ex) -- node(#4) {} ($(#5) + (#2:#3ex)$) -- (#5) % - ; % -} + \draw[->,#6,rounded corners] + (#1) -- +(#2:#3ex) -- node(#4) {} ($(#5) + (#2:#3ex)$) -- (#5) % + ; % + } \newcommand{\celltoangle}[5]{ - \path[draw] ($(#1)+(#2:#3ex)$) edge[celllr={0}{0},#5] ($(#1)+(#2:#4ex)$) ; % + \path[draw] ($(#1)+(#2:#3ex)$) edge[celllr={0}{0},#5] ($(#1)+(#2:#4ex)$) ; % } \newcommand{\pbkdefault}{1.4em} \newcommand{\pbkmargin}{1pt} \DeclareDocumentCommand{\pullbackk}{O{\pbkdefault} O{\pbkdefault} D(){2pt} m m m m}{% - \node[coordinate] (a) at (#4) {} ; % - \node[coordinate] (b) at (#5) {} ; % - \node[coordinate] (c) at (#6) {} ; % - \node[coordinate] (a') at ($(b)!#1!(a)$) {} ; % - \node[coordinate] (c') at ($(b)!#2!(c)$) {} ; % - \node[coordinate] (d) at (barycentric cs:a'=1,c'=1,b=-1) {} ; % - \node[coordinate] (aup) at ($(a')!#3!(d)$) {}; - \node[coordinate] (cup) at ($(c')!#3!(d)$) {}; - \path[#7] (aup) -- (d) -- (cup) ; % + \node[coordinate] (a) at (#4) {} ; % + \node[coordinate] (b) at (#5) {} ; % + \node[coordinate] (c) at (#6) {} ; % + \node[coordinate] (a') at ($(b)!#1!(a)$) {} ; % + \node[coordinate] (c') at ($(b)!#2!(c)$) {} ; % + \node[coordinate] (d) at (barycentric cs:a'=1,c'=1,b=-1) {} ; % + \node[coordinate] (aup) at ($(a')!#3!(d)$) {}; + \node[coordinate] (cup) at ($(c')!#3!(d)$) {}; + \path[#7] (aup) -- (d) -- (cup) ; % } \newcommand{\pullback}[5][\pbkdefault]{\pullbackk[#1][#1]{#2}{#3}{#4}{#5}} \newcommand{\pbk}[4][\pbkdefault]{% - \pullbackk[#1][#1]{#2}{#3}{#4}{draw,-} } + \pullbackk[#1][#1]{#2}{#3}{#4}{draw,-} } \newcommand{\stdpbk}{\pbk{m-2-1}{m-1-1}{m-1-2}} \newcommand{\stdpo}{\pbk{m-2-1}{m-2-2}{m-1-2}} \newcommand{\onepbk}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,densely dotted} } + \pullback[#1]{#2}{#3}{#4}{draw,densely dotted} } \newcommand{\ptwpbk}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,densely dotted} } + \pullback[#1]{#2}{#3}{#4}{draw,densely dotted} } \newcommand{\wpbk}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,dashed} } + \pullback[#1]{#2}{#3}{#4}{draw,dashed} } \newcommand{\pbkk}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{cell=0} } + \pullback[#1]{#2}{#3}{#4}{cell=0} } \newcommand{\laxpbk}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,->,cell=0} } + \pullback[#1]{#2}{#3}{#4}{draw,->,cell=0} } \newcommand{\oplaxpbk}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,<-,cell=0} } + \pullback[#1]{#2}{#3}{#4}{draw,<-,cell=0} } \newcommand{\poleftg}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,open triangle 45-} } + \pullback[#1]{#2}{#3}{#4}{draw,open triangle 45-} } \newcommand{\porightg}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,open triangle 45 reversed-} } + \pullback[#1]{#2}{#3}{#4}{draw,open triangle 45 reversed-} } \newcommand{\dpbk}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,-open triangle 45} } + \pullback[#1]{#2}{#3}{#4}{draw,-open triangle 45} } \newcommand{\dpbkrev}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,-open triangle 45 reversed} } + \pullback[#1]{#2}{#3}{#4}{draw,-open triangle 45 reversed} } \newcommand{\dpbkblack}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,-triangle 45} } + \pullback[#1]{#2}{#3}{#4}{draw,-triangle 45} } \newcommand{\dpbkblackrev}[4][\pbkdefault]{% - \pullback[#1]{#2}{#3}{#4}{draw,triangle 45-open triangle 45} } + \pullback[#1]{#2}{#3}{#4}{draw,triangle 45-open triangle 45} } \tikzset{shortenlr/.style 2 args={shorten <={#1 ex},shorten >={#2 ex}}} \tikzset{ - back/.style={densely dotted} +back/.style={densely dotted} } \tikzset{ - fore/.style 2 args={preaction={draw={white},-,line width=4pt,shorten <=#1cm,shorten >=#2cm}}, - fore/.default={0.2}{0.2} +fore/.style 2 args={preaction={draw={white},-,line width=4pt,shorten <=#1cm,shorten >=#2cm}}, +fore/.default={0.2}{0.2} } \tikzset{ - foretwo/.style={preaction={draw=white,-,line width=6pt}} +foretwo/.style={preaction={draw=white,-,line width=6pt}} } \tikzset{twocell/.style = {double equal sign distance,double,-implies,shorten <= .15cm,shorten >=.15cm,draw}} \tikzset{ - cell/.style = {double equal sign distance,double,-implies,shorten <= #1 cm,shorten >=#1 cm,draw} +cell/.style = {double equal sign distance,double,-implies,shorten <= #1 cm,shorten >=#1 cm,draw} } \tikzset{celllr/.style 2 args = {double equal sign distance,double,-implies,shorten <= #1 ex,shorten >=#2 ex,draw}} \tikzset{identity/.style = {double equal sign distance,double,-,draw}} \tikzset{iso/.style = {label={[below=0em,sloped]{$\scriptstyle -$}},label={[above=-.2em,sloped]{$\scriptstyle \sim$}}}} \tikzset{equi/.style = {label={[above=-.2em,sloped]{$\scriptstyle \sim$}}}} \tikzset{isotwo/.style = postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\node[inner sep=2pt,outer sep=0,above=-.2em,sloped] {$\scriptstyle \sim$} ;} -}}} + decorate, + decoration={ + markings, + mark=at position .5 with {\node[inner sep=2pt,outer sep=0,above=-.2em,sloped] {$\scriptstyle \sim$} ;} + }}} \tikzset{isor/.style = {labelr={\iso}}} \tikzset{isol/.style = {labell={\iso}}} \tikzset{isod/.style = {labeld={\iso}}} @@ -747,33 +747,33 @@ \tikzset{adj/.default={.1cm}{0cm}} \tikzset{iff/.style = {double equal sign distance,double,-implies,draw,shorten <= #1cm}} \tikzset{ - mod/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\draw[-] (0pt,-2pt) -- (0pt,2pt);} - }}}, - negate/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\node[transform shape] (tempnode) {$/$};} - }}}, - mapsto/.style={|->}, - otspam/.style={<-|}, - pro/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position #1 with {\draw[-,fill] (0pt,0pt) circle (1.5pt);} - }}}, - pro/.default={.5}, - glob/.style={postaction={ - decorate, - decoration={ - markings, - mark=at position .5 with {\draw[-,fill=white] (0pt,0pt) circle (1.5pt);} - }}} +mod/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with {\draw[-] (0pt,-2pt) -- (0pt,2pt);} +}}}, +negate/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with {\node[transform shape] (tempnode) {$/$};} +}}}, +mapsto/.style={|->}, +otspam/.style={<-|}, +pro/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position #1 with {\draw[-,fill] (0pt,0pt) circle (1.5pt);} +}}}, +pro/.default={.5}, +glob/.style={postaction={ +decorate, +decoration={ +markings, +mark=at position .5 with {\draw[-,fill=white] (0pt,0pt) circle (1.5pt);} +}}} } \newcommand{\idto}{\mathbin{\tikz[baseline] \draw[identity] (0pt,.5ex) -- (3ex,.5ex);}} @@ -781,17 +781,17 @@ \newcommand{\fibto}[1]{\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[fib,labelu={\scriptstyle #1}] (3ex,.5ex);}} \newcommand{\fibot}[1]{\mathbin{\tikz[baseline] \draw (3ex,.5ex) edge[fib,labelu={\scriptstyle #1}] (0pt,.5ex);}} \newcommand{\dfibto}{ - \mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib] (3ex,.5ex);}} + \mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib] (3ex,.5ex);}} \newcommand{\xdfibto}[1]{ - \mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib,labelu={\scriptstyle #1}] (3ex,.5ex);}} + \mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib,labelu={\scriptstyle #1}] (3ex,.5ex);}} \newcommand{\shortdfibto}{ - \mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib] (1.5ex,.5ex);}} + \mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[dfib] (1.5ex,.5ex);}} \newcommand{\shortdfibot}{ - \mathbin{\tikz[baseline] \draw (1.5ex,.5ex) edge[dfib] (0,.5ex);}} + \mathbin{\tikz[baseline] \draw (1.5ex,.5ex) edge[dfib] (0,.5ex);}} \newcommand{\xxto}[2]{\mathbin{\tikz[baseline] \draw (0pt,.5ex) edge[->,labelu={\scriptstyle #1},labelb={\scriptstyle #2}] (3ex,.5ex);}} \newcommand{\dfibot}[1][]{ - \mathbin{\tikz[baseline] \draw (3ex,.5ex) edge[dfib,labelu={\scriptstyle #1}] (0pt,.5ex);}} + \mathbin{\tikz[baseline] \draw (3ex,.5ex) edge[dfib,labelu={\scriptstyle #1}] (0pt,.5ex);}} \newcommand{\modto}{\mathbin{\tikz[baseline] \draw[->,mod] (0pt,.5ex) -- (3ex,.5ex);}} \newcommand{\proto}{\mathbin{\tikz[baseline] \draw[->,pro] (0pt,.5ex) -- (3ex,.5ex);}} \newcommand{\shortproto}{\mathbin{\tikz[baseline] \draw[->,pro] (0pt,.5ex) -- (1.5ex,.5ex);}} @@ -804,34 +804,34 @@ \newcommand{\cellule}[3][]{ % - \path (#3) +(#2:-.4cm) [twocell,#1] -- +(#2:.4cm) ; % + \path (#3) +(#2:-.4cm) [twocell,#1] -- +(#2:.4cm) ; % } \newcommand{\celluled}[2]{ % - \cellule[labell={#2}]{-90}{#1} % + \cellule[labell={#2}]{-90}{#1} % } \newcommand{\celluler}[2]{ % - \cellule[labelu={#2}]{0}{#1} % + \cellule[labelu={#2}]{0}{#1} % } \newcommand{\isopath}[2]{% - \path % - (#1) -- node[sloped] {$\iso$} (#2) ; % + \path % + (#1) -- node[sloped] {$\iso$} (#2) ; % } \newcommand{\adjtemplate}[6][1]{% - \begin{tikzpicture}[baseline=(m-1-1.base)] % - \matrix (m) [diag={1}{#1}, - column sep={#1 cm,between borders}]{ % - |[adj]| #2 % - \& #6 % - \& |[adj]| #3 \\ - } ; % - \path[->] % - (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % - (m-1-3.base west) edge[labeld={#5},bend left=15] (m-1-1.base east) % - ; % - \end{tikzpicture} % + \begin{tikzpicture}[baseline=(m-1-1.base)] % + \matrix (m) [diag={1}{#1}, + column sep={#1 cm,between borders}]{ % + |[adj]| #2 % + \& #6 % + \& |[adj]| #3 \\ + } ; % + \path[->] % + (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % + (m-1-3.base west) edge[labeld={#5},bend left=15] (m-1-1.base east) % + ; % + \end{tikzpicture} % } \newcommand{\adj}[5][1]{\adjtemplate[#1]{#2}{#3}{#4}{#5}{\bot}} @@ -840,107 +840,107 @@ \newcommand{\catiso}[5][1]{\adjtemplate[#1]{#2}{#3}{#4}{#5}{\cong}} \newcommand{\adjunction}[4]{% - \path[->] % - (#1) edge[twou={#3},bend left=15] (#2) % - (#2) edge[twod={#4},bend left=15] (#1) % - ; % - \path (u) -- node[pos=.5,sloped] {$\dashv$} (d) ; % -} + \path[->] % + (#1) edge[twou={#3},bend left=15] (#2) % + (#2) edge[twod={#4},bend left=15] (#1) % + ; % + \path (u) -- node[pos=.5,sloped] {$\dashv$} (d) ; % + } \newcommand{\ladjunction}[4]{% - \path[->] % - (#1) edge[twou={#3},bend left=15] (#2) % - (#2) edge[twod={#4},bend left=15] (#1) % - ; % - \path (u) -- node[pos=.5,sloped] {$\vdash$} (d) ; % -} + \path[->] % + (#1) edge[twou={#3},bend left=15] (#2) % + (#2) edge[twod={#4},bend left=15] (#1) % + ; % + \path (u) -- node[pos=.5,sloped] {$\vdash$} (d) ; % + } \newcommand{\adjs}[8][1]{% - \begin{tikzpicture} % - \matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ % - |[adj]| #2 % - \& \bot % - \& |[adj]| #3 - \& \bot % - \& |[adj]| #6 \\ - } ; % - \path[->] % - (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % - (m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) % - (m-1-3.north east) edge[labelu={#7},bend left=15] (m-1-5.north west) % - (m-1-5.south west) edge[labeld={#8},bend left=15] (m-1-3.south east) % - ; % - \end{tikzpicture} % + \begin{tikzpicture} % + \matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ % + |[adj]| #2 % + \& \bot % + \& |[adj]| #3 + \& \bot % + \& |[adj]| #6 \\ + } ; % + \path[->] % + (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % + (m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) % + (m-1-3.north east) edge[labelu={#7},bend left=15] (m-1-5.north west) % + (m-1-5.south west) edge[labeld={#8},bend left=15] (m-1-3.south east) % + ; % + \end{tikzpicture} % } \newcommand{\coadjs}[8][1]{% - \begin{tikzpicture} % - \matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ % - |[adj]| #2 % - \& \top % - \& |[adj]| #3 - \& \top % - \& |[adj]| #6 \\ - } ; % - \path[->] % - (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % - (m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) % - (m-1-3.north east) edge[labelu={#7},bend left=15] (m-1-5.north west) % - (m-1-5.south west) edge[labeld={#8},bend left=15] (m-1-3.south east) % - ; % - \end{tikzpicture} % + \begin{tikzpicture} % + \matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ % + |[adj]| #2 % + \& \top % + \& |[adj]| #3 + \& \top % + \& |[adj]| #6 \\ + } ; % + \path[->] % + (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % + (m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) % + (m-1-3.north east) edge[labelu={#7},bend left=15] (m-1-5.north west) % + (m-1-5.south west) edge[labeld={#8},bend left=15] (m-1-3.south east) % + ; % + \end{tikzpicture} % } \newcommand{\retr}[5][1]{% - \begin{tikzpicture} % - \matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ % - |[anchor=east,text height=.1cm,text depth=-.1cm]| #2 % - \& |[anchor=center]| \triangleleft % - \& |[anchor=west,text height=.1cm,text depth=-.1cm]| #3 \\ - } ; % - \path[->] % - (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % - (m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) % - ; % - \end{tikzpicture} % + \begin{tikzpicture} % + \matrix (m) [diag={1}{#1},column sep={#1 cm,between borders}]{ % + |[anchor=east,text height=.1cm,text depth=-.1cm]| #2 % + \& |[anchor=center]| \triangleleft % + \& |[anchor=west,text height=.1cm,text depth=-.1cm]| #3 \\ + } ; % + \path[->] % + (m-1-1.north east) edge[labelu={#4},bend left=15] (m-1-3.north west) % + (m-1-3.south west) edge[labeld={#5},bend left=15] (m-1-1.south east) % + ; % + \end{tikzpicture} % } \newcommand{\doublecell}[9]{ - \diag{% - |(X)| {#1} \& |(Y)| {#2} \\ % - |(U)| {#3} \& |(V)| {#4} % - }{% - (X) edge[labelu={#5}] (Y) % - edge[pro,twol={#6}] (U) % - (Y) edge[pro,twor={#7}] (V) % - (U) edge[labeld={#8}] (V) % - (l) edge[cell=.4,labelu={\scriptstyle #9}] (r) % - } + \diag{% + |(X)| {#1} \& |(Y)| {#2} \\ % + |(U)| {#3} \& |(V)| {#4} % + }{% + (X) edge[labelu={#5}] (Y) % + edge[pro,twol={#6}] (U) % + (Y) edge[pro,twor={#7}] (V) % + (U) edge[labeld={#8}] (V) % + (l) edge[cell=.4,labelu={\scriptstyle #9}] (r) % + } } \newcommand{\doublecellpro}[9]{ - \diag{% - |(X)| {#1} \& |(Y)| {#2} \\ % - |(U)| {#3} \& |(V)| {#4} % - }{% - (X) edge[labelu={#5}] (Y) % - edge[pro,twol={#6}] (U) % - (Y) edge[pro,twor={#7}] (V) % - (U) edge[labeld={#8}] (V) % - (l) edge[cell=.4,labelu={\scriptstyle #9}] (r) % - } + \diag{% + |(X)| {#1} \& |(Y)| {#2} \\ % + |(U)| {#3} \& |(V)| {#4} % + }{% + (X) edge[labelu={#5}] (Y) % + edge[pro,twol={#6}] (U) % + (Y) edge[pro,twor={#7}] (V) % + (U) edge[labeld={#8}] (V) % + (l) edge[cell=.4,labelu={\scriptstyle #9}] (r) % + } } \newcommand{\vdoublecell}[9]{ - \diag{% - |(X)| {#1} \& |(Y)| {#2} \\ % - |(U)| {#3} \& |(V)| {#4} % - }{% - (X) edge[twou={#5}] (Y) % - edge[labell={#6}] (U) % - (Y) edge[labelr={#7}] (V) % - (U) edge[twod={#8}] (V) % - (u) edge[cell=.4,labelr={\scriptstyle #9}] (d) % - } + \diag{% + |(X)| {#1} \& |(Y)| {#2} \\ % + |(U)| {#3} \& |(V)| {#4} % + }{% + (X) edge[twou={#5}] (Y) % + edge[labell={#6}] (U) % + (Y) edge[labelr={#7}] (V) % + (U) edge[twod={#8}] (V) % + (u) edge[cell=.4,labelr={\scriptstyle #9}] (d) % + } } \newcommand{\Vdots}{|[anchor=center,text height=.1cm]| \vdots} @@ -951,19 +951,19 @@ \newcommand{\mkdots}[2]{ \path - (#1) -- - node[pos=.4] {.} - node[pos=.5] {.} - node[pos=.6] {.} - (#2) - ;} + (#1) -- + node[pos=.4] {.} + node[pos=.5] {.} + node[pos=.6] {.} + (#2) + ;} \newcommand{\mkdotsshrink}[2]{ \path - (#1) -- - node[pos=.3] {.} - node[pos=.5] {.} - node[pos=.7] {.} - (#2) - ;} + (#1) -- + node[pos=.3] {.} + node[pos=.5] {.} + node[pos=.7] {.} + (#2) + ;} \endinput