% (find-LATEX "2020dialectica.tex")
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% https://golem.ph.utexas.edu/category/2020/07/linear_logic_flavoured_composi.html

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\def\ton#1{\overset{#1}{\to}}
% (find-symbolspage 84 "Table 161: stix Negated Arrows")
% (find-symbolstext 84 "Table 161: stix Negated Arrows")
\def\cutot                {\nvleftarrow}
\def\cutotn#1{\overset{#1}{\nvleftarrow}}

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

Notes on Valeria de Paiva's PhD thesis: ``The Dialectica Categories'' (1991)

\url{https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf}

\url{http://vcvpaiva.github.io/}

\ssk

These notes are at:

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

\url{http://angg.twu.net/math-b.html\#notes-on-notation-2020}

}


% (find-books "__cats/__cats.el" "depaiva")
% (find-depaivaphdpage (+ 6  3) "1. The Dialectica Categories DC")







\section*{1. The Dialectica Categories DC}

A morphism $(f,F): (U\cutotn{α}X) \to (V\cutotn{β}Y)$ is a pair of
arrows $f$ and $F$ like this

%D diagram sec.1.1_morphism-1
%D 2Dx     100 +35 +20 +20 +35 +20 +20
%D 2D  100 A0  B0      B1  C0      C1
%D 2D
%D 2D  +20         BM          CM
%D 2D
%D 2D  +20 A1  B2      B3  C2      C3
%D 2D
%D ren A0 A1 ==> (U\cutotn{α}X) (V\cutotn{β}Y)
%D ren B0 B1 B2 B3 BM ==> U X V Y \vdpnearrow
%D ren C0 C1 C2 C3 CM ==> U X V Y U×Y
%D
%D (( A0 A1 -> .plabel= l (f,F)
%D
%D    B1 B0 <- .plabel= a α
%D    B1 B0 <- .plabel= m |
%D    B0 B2 -> .plabel= l f
%D    B2 B3 -> .plabel= a β
%D    B2 B3 -> .plabel= m |
%D    BM place
%D    BM relplace 10 0 F
%D
%D    C0 C2 -> .plabel= l f
%D    CM C1 -> .plabel= r F
%D    C3 place
%D ))
%D enddiagram
%D
$$\pu
  \diag{sec.1.1_morphism-1}
$$

such that the inclusion $A' \diagxyto/>-->/<250> B'$ below exists:

%D diagram sec.1.1_morphism-2
%D 2Dx     100 +20  +40  +50 +20  +40
%D 2D  100 A0 >---> A1   B0 >---> B1 
%D 2D      |       ^     |       ^   
%D 2D  +20 A2     /      B2     /    
%D 2D      v  \  /       v  \  /     
%D 2D  +20 |   A3        |   B3      
%D 2D      v  /  \       v  /  \     
%D 2D  +20 A4     \      B4     \    
%D 2D      |       v     |       v   
%D 2D  +20 A5 >---> A6   B5 >---> B6 
%D 2D
%D ren A0  A1  B0  B1 ==> A  U×X  \setofst{(u,x)}{uαx}      U×X
%D ren A2      B2     ==> A'      \setofst{(u,y)}{uαF(v,y)}
%D ren   A3      B3   ==>  U×Y                           U×Y
%D ren A4      B4     ==> B'      \setofst{(u,y)}{f(u)βy}
%D ren A5  A6  B5  B6 ==> B  V×Y  \setofst{(x,y)}{vβy}      V×Y
%D
%D (( A0 A1 >-> .plabel= a α
%D    A0 A2 <-
%D    A2 A3 >-> .plabel= a α'
%D    A1 A3 <-  .plabel= r 〈π_1,F〉
%D    A2 A4 >-->
%D    A3 A4 <-< .plabel= b β'
%D    A3 A6 ->  .plabel= r f×\id
%D    A4 A5 ->
%D    A5 A6 >-> .plabel= b β
%D
%D    B0 B1 >->
%D    B0 B2 <-
%D    B2 B3 >->
%D    B1 B3 <-  .plabel= r (u,y)↦(u,F(u,y))
%D    B2 B4 >--> .plabel= l \sm{∀(u,y).\\uαF(v,y)\\→f(u)βy}\;\;
%D    B3 B4 <-<
%D    B3 B6 ->  .plabel= r (u,y)↦(f(u),y)
%D    B4 B5 ->
%D    B5 B6 >->
%D ))
%D enddiagram
%D
$$\pu
  \diag{sec.1.1_morphism-2}
$$



\end{document}

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