% (find-LATEX "2020macdonaldsobral.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020macdonaldsobral.tex" :end))
% (defun d () (interactive) (find-pdf-page      "~/LATEX/2020macdonaldsobral.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020macdonaldsobral.pdf"))
% (defun e () (interactive) (find-LATEX "2020macdonaldsobral.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020macdonaldsobral"))
% (defun v () (interactive) (find-2a '(e) '(d)) (g))
% (find-pdf-page   "~/LATEX/2020macdonaldsobral.pdf")
% (find-sh0 "cp -v  ~/LATEX/2020macdonaldsobral.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2020macdonaldsobral.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2020macdonaldsobral.pdf
%               file:///tmp/2020macdonaldsobral.pdf
%           file:///tmp/pen/2020macdonaldsobral.pdf
% http://angg.twu.net/LATEX/2020macdonaldsobral.pdf
% (find-LATEX "2019.mk")

% «.title»			(to "title")
% «.monads»			(to "monads")
% «.EM-construction»		(to "EM-construction")
% «.EM-adjunction»		(to "EM-adjunction")
% «.kleisli-construction»	(to "kleisli-construction")
% «.kleisli-adjunction»		(to "kleisli-adjunction")

\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
%\usepackage{colorweb}                 % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy        % For 2D diagrams ("%D" lines)
\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx15}               % (find-LATEX "edrx15.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
% (find-es "tex" "geometry")
\begin{document}

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")



% «title»  (to ".title")

{\setlength{\parindent}{0em}
\footnotesize

Notes on John MacDonald and Manuela Sobral's ``Aspects of Monads'':

\url{https://doi.org/10.1017/CBO9781107340985.008}

a chapter of ``Categorical Foundations: Special Topics in Order,
Topology, Algebra, and Sheaf Theory'', edited by Maria Cristina
Pedicchio and Walter Tholen, Cambridge, 2003.

\ssk

These notes are at:

\url{http://angg.twu.net/LATEX/2020macdonaldsobral.pdf}

}


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% |_|  |_|\___/|_| |_|\__,_|\__,_|___/
%                                     
% «monads»  (to ".monads")

\section*{1.2. Monads}

% (find-books "__cats/__cats.el" "macdonald-sobral")
% (find-mcdsobralpage (+ -212 216) "1.2. Monads")
% (find-mcdsobraltext (+ -212 216) "1.2. Monads")
% (find-mcdsobralpage (+ -212 217) "Proposition. Any adjunction")
% (find-mcdsobraltext (+ -212 217) "Proposition. Any adjunction")
% (find-symbolspage 70 "\\rightharpoonup")
% (find-symbolstext 70 "\\rightharpoonup")

(Page 216):

A monad on a category $𝐛X$ is a system $(T,η,μ)$...

(Page 217):

Proposition: any adjunction $(F,G,η,ε): 𝐛X \rightharpoonup 𝐛A$
determines a monad...

%D diagram ??
%D 2Dx     100 +20 +20 +20 +20 +20 +20 +20
%D 2D  100 A0  B0  B1  D0  E0 
%D 2D                         
%D 2D  +20 A1  B2  B3  D1  E1  F0  F1  F2
%D 2D                         
%D 2D  +20     C0  C1  D2  E2  F3  F4  F5
%D 2D
%D ren A0 A1       ==> FGA A
%D ren B0 B1 B2 B3 ==> FX X A GA
%D ren C0 C1       ==> 𝐛A 𝐛X
%D ren D0 D1 D2    ==> X GFX GFGFX
%D ren E0 E1 E2    ==> 1 T T^2
%D ren F0 F1 F2    ==> T T^2 T^3
%D ren F3 F4 F5    ==> T^2 T T^2
%D
%D (( A0 A1 ->  .plabel= l εA
%D    B0 B1 <-|
%D    B0 B2 ->
%D    B1 B3 ->
%D    B2 B3 |->
%D    C0 C1 <- sl^ .plabel= a F
%D    C0 C1 -> sl_ .plabel= b G
%D    D0 D1 ->     .plabel= r ηX
%D    D1 D2 <-     .plabel= r GεFX
%D    E0 E1 ->     .plabel= r η
%D    E1 E2 <-     .plabel= r μ
%D    F0 F1 ->     .plabel= a Tη
%D    F1 F2 <-     .plabel= a Tμ
%D    F0 F3 ->     .plabel= l ηT
%D    F0 F4 ->     .plabel= m 1_𝐛X
%D    F1 F4 ->     .plabel= r μ
%D    F2 F5 ->     .plabel= r μT
%D    F3 F4 ->     .plabel= b μ
%D    F4 F5 <-     .plabel= b μ
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

%  _____ __  __    ____ 
% | ____|  \/  |  / ___|
% |  _| | |\/| | | |    
% | |___| |  | | | |___ 
% |_____|_|  |_|  \____|
%                       
% «EM-construction»  (to ".EM-construction")
% (mdsp 1 "EM-construction")
% (mds    "EM-construction")

\section*{1.3. The Eilenberg-Moore Construction}

(Page 217):

% (find-mcdsobralpage (+ -212 217) "1.3. The Eilenberg-Moore construction")
% (find-mcdsobraltext (+ -212 217) "1.3. The Eilenberg-Moore construction")

...category of T-algebras which will be denoted by $𝐛X^T$.

%D diagram ??
%D 2Dx     100 +20 +20 +20
%D 2D  100 A0  B0  B1  B2
%D 2D  +10 A0
%D 2D  +10         B3  B4
%D 2D
%D 2D  +20 A1      C1  C2
%D 2D
%D 2D  +20 A2      C3  C4
%D 2D
%D 2D  +20 A3      D0
%D 2D
%D ren A0 A1 A2 A3 ==> (X,ξ) (X,ξ) (Y,Θ) 𝐛X^T
%D ren B0 B1 B2    ==> X TX T^2X
%D ren    B3 B4    ==>    X   TX
%D ren    C1 C2    ==>    X   TX
%D ren    C3 C4    ==>    Y   TY
%D ren    D0       ==>   𝐛X
%D
%D (( A0 place
%D    A1 A2 -> .plabel= l f
%D    A3 place
%D
%D    B0 B1 -> .plabel= a ηX
%D    B1 B2 <- .plabel= a μX
%D    B0 B3 -> .plabel= l 1_X
%D    B1 B3 -> .plabel= r ξ
%D    B2 B4 -> .plabel= r Tξ
%D    B3 B4 <- .plabel= a ξ
%D
%D    C1 C2 <- .plabel= a ξ
%D    C1 C3 -> .plabel= l f
%D    C2 C4 -> .plabel= r Tf
%D    C3 C4 <- .plabel= a θ
%D    D0 xy+= 10 0
%D    D0 place
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

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% | ____|  \/  |   __ _  __| |(_)
% |  _| | |\/| |  / _` |/ _` || |
% | |___| |  | | | (_| | (_| || |
% |_____|_|  |_|  \__,_|\__,_|/ |
%                           |__/ 
%
% «EM-adjunction»  (to ".EM-adjunction")

\subsection*{The Eilenberg-Moore adjunction}

(Page 218):

Proposition: For a monad $T=(T,η,μ)$ on $𝐛X$ there is a free-forgetful
adjunction
%
$$𝐛X^T \two/->`<-/<250>^{G^T}_{F^T} 𝐛X$$
%
which induces the monad $T$ in $𝐛X$.


%D diagram ??
%D 2Dx     100 +35 +30 +20 +20 +30 +25
%D 2D  100     A0  A1              E1
%D 2D
%D 2D  +20 C0  A2  A3      D1  E2  E3
%D 2D
%D 2D  +20 C1  A4  A5  D2  D3  E4  E5
%D 2D
%D 2D  +20     A6  A7
%D 2D
%D 2D  +20     B0  B1
%D 2D
%D ren A0 A1 ==> (TX',μX') X'
%D ren A2 A3 ==>  (TX,μX) TX
%D ren A4 A5 ==>   (Y,θ)   Y
%D ren A6 A7 ==>  (Y',θ') Y'
%D ren B0 B1 ==> 𝐛X^T 𝐛X
%D ren C0 C1 ==> (TX,μX) (X,ξ)
%D ren    D1 ==>   TX
%D ren D2 D3 ==> Y TY
%D ren    E1 ==>         X
%D ren E2 E3 ==> (TX,μ) TX
%D ren E4 E5 ==>  (Y,θ)  Y
%D
%D (( A0 A1 <-|
%D    A0 A2  -> .plabel= l Tf
%D    A1 A3  -> .plabel= r  f
%D    A2 A3 <-|
%D    A4 A5 |->
%D    A4 A6  ->
%D    A5 A7  ->
%D    A6 A7 |->
%D    B0 B1  <- sl^ .plabel= a F^T
%D    B0 B1  -> sl_ .plabel= b G^T
%D
%D    C0 C1  -> .plabel= l \sm{ε^T(X,ξ):=\\ξ}
%D
%D    D1 D2  -> .plabel= a \ovl{f}
%D    D1 D3  -> .plabel= r Tf
%D    D2 D3  <- .plabel= b θ
%D
%D    E1 E3  -> .plabel= r \sm{η^TX:=\\ηX\\\text{(univ)}}
%D    E2 E3 |->
%D    E2 E4  -> .plabel= l \sm{∃!\ovl{f}:=\\θ·Tf}
%D    E3 E5  -> .plabel= r \ovl{f}
%D    E4 E5 |->
%D    E1 E5  -> .slide= 35pt .plabel= r ∀f
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

\newpage

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%                                
% «kleisli-construction»  (to ".kleisli-construction")
% (find-mcdsobralpage (+ -212 220) "1.6. The Kleisli construction")
% (find-mcdsobraltext (+ -212 220) "1.6. The Kleisli construction")
% (mdsp 4 "kleisli-construction")
% (mds    "kleisli-construction")

\subsection*{1.6. The Kleisli construction}

(Page 220):

%D diagram ??
%D 2Dx     100 +20 +20 +20 +35
%D 2D  100 A0  F0          R0
%D 2D
%D 2D  +20 B0  G0          
%D 2D  +10                 R1
%D 2D  +10 B1      G1
%D 2D
%D 2D  +20 C0  H0
%D 2D  +10                 R2
%D 2D  +10 C1      H1
%D 2D
%D 2D  +20 D0  I0
%D 2D  
%D 2D  +20 D1      I1      R3
%D 2D
%D 2D  +20 D2      I3  I2
%D 2D
%D 2D  +20 E0  J0
%D 2D
%D ren A0 F0       ==> X X
%D ren B0 B1 G0 G1 ==> X Y X TY
%D ren C0 C1 H0 H1 ==> X X X TX
%D ren D0 D1 D2 I0 I1 I2 I3 ==> X Y Z X TY T^2X TZ
%D ren E0 J0       ==> 𝐛X_T 𝐛X
%D
%D (( A0 place
%D    B0 B1 -> .plabel= l f
%D    C0 C1 -> .plabel= l \id_X
%D    D0 D1 -> .plabel= l f
%D    D1 D2 -> .plabel= l g
%D    E0 place
%D
%D    F0 place
%D    G0 G1 -> .plabel= r f
%D    H0 H1 -> .plabel= r η_X
%D    I0 I1 -> .plabel= r f
%D    I1 I2 -> .plabel= r Tg
%D    I2 I3 -> .plabel= b μ_Z
%D    J0 xy+= 10 0 place
%D
%D    R0 .tex= \defobjs place
%D    R1 .tex= \defhoms place
%D    R2 .tex= \defids  place
%D    R3 .tex= \defcomp place
%D ))
%D enddiagram
%D
$$\pu
  \def\MLL#1#2#3#4{\begin{array}{l} #1\\ #2 \\ #3 \\ #4 \end{array}} 
  \def\ML#1#2{\begin{array}{l} #1\\ #2 \end{array}} 
  \def\MC#1#2{\begin{array}{l} #1\\ #2 \end{array}} 
  \def\MR#1#2{\begin{array}{r} #1\\ #2 \end{array}} 
  \def\defobjs{\begin{array}{l}
                 \Objs(𝐛X_T) := \\
                 \Objs(𝐛X)      \\
               \end{array}
              }
  \def\defhoms{\begin{array}{l}
                 \Hom_{𝐛X_T}(X,Y) :=       \\
                 \Hom_{𝐛X}(X,TY)           \\
                 f:X→Y \text{ in } 𝐛X_T := \\
                 f:X→TY \text{ in } 𝐛X     \\
               \end{array}
              }
  \def\defids {\begin{array}{l}
                 \id_{𝐛X_T}(X) := \\
                 η_X              \\
               \end{array}
              }
  \def\defcomp{\begin{array}{c}
                      g·f \text{ in } 𝐛X_T := \\
                 μ_Z·Tg·f \text{ in } 𝐛X      \\
               \end{array}
              }
  \diag{??}
$$

\newpage

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% «kleisli-adjunction»  (to ".kleisli-adjunction")
% (mdsp 4 "kleisli-adjunction")
% (mds    "kleisli-adjunction")
% (find-mcdsobralpage (+ -212 221) "Kleisli adjunction")
% (find-mcdsobraltext (+ -212 221) "Kleisli adjunction")

\section*{The Kleisli adjunction}

(Page 221):

For a given monad $T$ there exists the Kleisli adjunction...


\end{document}

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% <make>

 (eepitch-shell)
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# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020macdonaldsobral veryclean
make -f 2019.mk STEM=2020macdonaldsobral pdf

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