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% This file: (find-LATEX "2019J-ops-kan.tex")
% See: (find-LATEX "2020J-ops-new.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2019J-ops-kan.tex" :end))
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% «.kan-extensions» (to "kan-extensions")
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% (jonp 36 "kan-extensions")
% (jop "kan-extensions")
% (favp 45 "kan-extensions")
% (fav "kan-extensions")
\subsection{Kan extensions}
\label {kan-extensions}
\def\Lan{\text{Lan}}
\def\Ran{\text{Ran}}
\def\sfC{\mathsf{C}}
\def\sfD{\mathsf{D}}
\def\sfE{\mathsf{E}}
% (find-books "__cats/__cats.el" "riehl")
% (find-riehlccpage (+ 18 44) "1.7. The 2-category of categories")
% (find-riehlcctext (+ 18 44) "1.7. The 2-category of categories")
% (find-riehlccpage (+ 18 45) "Lemma 1.7.4 (horizontal composition)")
% (find-riehlcctext (+ 18 45) "Lemma 1.7.4 (horizontal composition)")
% (find-riehlccpage (+ 18 46) "whiskering")
% (find-riehlcctext (+ 18 46) "whiskering")
% (find-riehlccpage (+ 18 189) "6. All Concepts are Kan Extensions")
% (find-riehlccpage (+ 18 190) "6.1. Kan extensions")
% (find-riehlccpage (+ 18 190) "Dually, a right Kan")
% (find-riehlcctext (+ 18 190) "Dually, a right Kan")
In \cite{Riehl}, sec.6.1, right Kan extensions are explained using the
two diagrams below. The notation of cells is explained in sec.1.7 of
the book, and modulo the types --- that can be inferred from the
diagrams --- a right Kan extension of $K$ along $K$ is a pair $(\Ran_K
F,ε)$ such that for all $(G,α)$ there is a unique $β$ making
everything commute.
%
%D diagram riehl-ran-1
%D 2Dx 100 +40 +40
%D 2D 100 A0 ---> A2
%D 2D -> ->
%D 2D +40 A1
%D 2D
%D ren A0 A1 A2 ==> \mathsf{C} \mathsf{D} \mathsf{E}
%D
%D (( A0 A2 -> .plabel= a F
%D A0 A1 -> .plabel= l K
%D A1 A2 -> .plabel= r G .curve= _25pt
%D A1 A2 varrownodes nil 17 nil <= .slide= -5pt .plabel= r δ
%D ))
%D enddiagram
%D
%D diagram riehl-ran-factored
%D 2Dx 100 +40 +40
%D 2D 100 A0 ---> A2
%D 2D -> -> ^
%D 2D +40 A1 -/
%D 2D
%D ren A0 A1 A2 ==> \mathsf{C} \mathsf{D} \mathsf{E}
%D
%D (( A0 A2 -> .plabel= a F
%D A0 A1 -> .plabel= l K
%D A1 A2 -> .plabel= m \Ran_KF
%D A1 A2 -> .plabel= r G .curve= _25pt
%D A0 A2 varrownodes 35 17 nil <= .plabel= l ε
%D A1 A2 varrownodes 20 17 nil <= .slide= 5pt .plabel= r β
%D ))
%D enddiagram
%D
$$\pu
\diag{riehl-ran-1}
\quad
\diag{riehl-ran-factored}
$$
If we specialize $\sfE$ to $\Set$ and do some renamings, the diagram
becomes:
%
%D diagram my-ran-1
%D 2Dx 100 +40 +40
%D 2D 100 A0 A2
%D 2D
%D 2D +40 A1
%D 2D
%D ren A0 A1 A2 ==> \catA \catB \Set
%D
%D (( A0 A2 -> .plabel= a D
%D A0 A1 -> .plabel= l f
%D A1 A2 -> .plabel= r C .curve= _25pt
%D A1 A2 varrownodes nil 17 nil <= .slide= -5pt .plabel= r α
%D ))
%D enddiagram
%D
%D diagram my-ran-2
%D 2Dx 100 +40 +40
%D 2D 100 A0 A2
%D 2D
%D 2D +40 A1
%D 2D
%D ren A0 A1 A2 ==> \catA \catB \Set
%D
%D (( A0 A2 -> .plabel= a D
%D A0 A1 -> .plabel= l f
%D A1 A2 -> .plabel= m \Ran_fD
%D A1 A2 -> .plabel= r C .curve= _25pt
%D A0 A2 varrownodes 35 17 nil <= .plabel= l ε
%D A1 A2 varrownodes 20 17 nil <= .slide= 5pt .plabel= r β
%D ))
%D enddiagram
%D
$$\pu
\diag{my-ran-1}
\quad
\diag{my-ran-2}
$$
%
and if we change its {\sl shape} to stress that $ε$ ``looks like'' a
counit map and $\Ran_f$ ``looks like'' the right adjoint to the
functor $f^*$, we get this:
%
%D diagram geo-morph
%D 2Dx 100 +30 +35 +30
%D 2D 100 L0 C0 C1 R1
%D 2D
%D 2D +35 L2 C2 C3 R3
%D 2D
%D 2D +20 C4 C5
%D 2D
%D 2D +20 C6 C7
%D 2D
%D ren C0 C1 C2 C3 C4 C5 ==> f^*C C D \Ran_fD \Set^\catA \Set^\catB
%D ren C6 C7 ==> \catA \catB
%D ren L0 L2 ==> f^*\Ran_fD D
%D ren R1 R3 ==> C \Ran_ff^*C
%D
%D (( C0 C1 <-|
%D C0 C2 -> .plabel= l \sm{β^\fl\\α}
%D C1 C3 -> .plabel= r \sm{β\\α^♯}
%D C2 C3 |->
%D C0 C3 harrownodes nil 20 nil <-| sl^
%D C0 C3 harrownodes nil 20 nil |-> sl_
%D
%D C4 C5 <- sl^ .plabel= a f^*
%D C4 C5 -> sl_ .plabel= b \Ran_f
%D
%D C6 C7 -> .plabel= a f
%D L0 L2 -> .plabel= l ε
%D R1 R3 -> .plabel= r d
%D ))
%D enddiagram
%D
$$\pu
\diag{geo-morph}
$$
When the categories $\catA$ and $\catB$ are finite posets we get:
\begin{itemize}
\item $\Set^\catA$ and $\Set^\catB$ are toposes (we saw this in
sec.\ref{Set-PA}),
%
% (jopp 23 "Set-PA")
% (joe "Set-PA")
\item the functor $f^*$ is ``precomposition with $f$'', in this sense:
if $C$ is an object of $\Set^B$ and $A∈\catA$ then $(f^*C)(A)$ is
$C(f(A))$,
\item the left and right Kan extensions $\Lan_f$ and $\Ran_f$ and can be
defined and calculated by the formulas in sec.6.2 of \cite{Riehl},
%
% (elep 7 "elephant-A4.1.4")
% (ele "elephant-A4.1.4")
% (find-books "__cats/__cats.el" "riehl")
% (find-riehlccpage (+ 18 193) "6.2. A formula for Kan extensions")
\item we have adjunctions $\Lan_f ⊣ f^* ⊣ \Ran_f$, and so the structure
$(\Lan_f ⊣ f^* ⊣ \Ran_f)$ can be seen as an essential geometric
morphism $f:\Set^\catA → \Set^\catB$ (\cite{EA}, A4.1.4); as $f^*$
is a right adjoint it preserves limits (\cite{Riehl}, sec.4.5,
and \cite{Awodey}, sec.9.6), and so $(f^* ⊣ \Ran_f)$ is a geometric
morphism $f:\Set^\catA → \Set^\catB$. We usually rename $(\Lan_f ⊣
f^* ⊣ \Ran_f)$ to $(f^! ⊣ f^* ⊣ f_*)$
\item when $f:\catA→\catB$ is something very simple we can find $\Ran_f D$
``by hand'' --- for example, in the example below, discussed
in \cite{OchsACT2019}:
%
% (oxap 5 "fig:internal-gms")
% (oxa "fig:internal-gms")
%
%R sesw = {[" w"]="↙", [" e"]="↘"}
%R
%R local zcB, zpBC, zpBRD
%R = 3/ 1 \, 3/ C_1 \, 3/ !Dt \
%R | w e | | w e | | w e |
%R | 2 3 | |C_2 C_3 | |D_2 D_3 |
%R | e w e | | e w e | | e w e |
%R | 4 5 | | C_4 C_5| | D_4 D_5|
%R | e w | | e w | | e w |
%R \ 6 / \ C_6 / \ 1 /
%R
%R local zpBRLC
%R = 3/ !Ct \
%R | w e |
%R |C_2 C_3 |
%R | e w e |
%R | C_4 C_5|
%R | e w |
%R \ 1 /
%R
%R local zcA, zpALC, zpAD
%R = 3/ 2 3 \, 3/C_2 C_3 \, 3/D_2 D_3 \
%R | e w e | | e w e | | e w e |
%R \ 4 5 / \ C_4 C_5/ \ D_4 D_5/
%R
%R zcB :tozmp({def="zcB", scale="7pt", meta="s p"}):addcells(sesw):output()
%R zpBC :tozmp({def="zpBC", scale="7pt", meta="s p"}):addcells(sesw):output()
%R zpBRD :tozmp({def="zpBRD" , scale="7pt", meta="s p"}):addcells(sesw):output()
%R zpBRLC:tozmp({def="zpBRLC", scale="7pt", meta="s p"}):addcells(sesw):output()
%R zcA :tozmp({def="zcA", scale="7pt", meta="s p"}):addcells(sesw):output()
%R zpALC :tozmp({def="zpALC", scale="7pt", meta="s p"}):addcells(sesw):output()
%R zpAD :tozmp({def="zpAD", scale="7pt", meta="s p"}):addcells(sesw):output()
%
%D diagram internal-zgm-particular-case
%D 2Dx 100 +50 +60 +55
%D 2D 100 A1 B1 <-| B2 C1
%D 2D | | | |
%D 2D | | <-> | |
%D 2D v v v v
%D 2D +50 A2 B3 |-> B4 C2
%D 2D
%D 2D +30 D1 <=> D2
%D 2D
%D 2D +20 E1 --> E2
%D 2D
%D 2D +30 F1 F2
%D 2D
%D ren A1 B1 B2 C1 ==> \zpALC \zpALC \zpBC \zpBC
%D ren A2 B3 B4 C2 ==> \zpAD \zpAD \zpBRD \zpBRLC
%D ren D1 D2 ==> \Set^\catA \Set^\catB
%D ren E1 E2 ==> \Set^\catA \Set^\catB
%D ren F1 F2 ==> \catA \catB
%D ren F1 F2 ==> \zcA \zcB
%D
%D (( A1 A2 -> .plabel= l εD
%D C1 C2 -> .plabel= r ηC
%D
%D B1 B2 <-|
%D B1 B3 -> B2 B4 ->
%D B3 B4 |->
%D B1 B4 harrownodes nil 20 nil <->
%D
%D D1 D2 <- sl^ .plabel= a f^*
%D D1 D2 -> sl_ .plabel= b f_*
%D E1 E2 -> .plabel= a f
%D
%D F1 F2 -> .plabel= a f
%D ))
%D enddiagram
%D
%
$$\pu
\def\Ct{C_2 {×_{C_4}} C_3}
\def\Dt{D_2 {×_{D_4}} D_3}
\diag{internal-zgm-particular-case}
$$
\end{itemize}
\bsk
Every situation in which the category $\catB$ is a $(P,A)$ and the
category $\catA$ is the full subcategory of $(P,A)$ whose objects are
$P∖Q$ yields a situation like the one in the diagram above, in which
the maps $εD$ are isos, the geometric morphism $f$ is an ``inclusion''
and the functor that takes each $C$ to $f_*f^*C$ is a sheafification
functor. A diagram with an example fully worked out will be included
in the next version of this paper at the Arxiv.
% (find-books "__cats/__cats.el" "riehl")
% (find-riehlccpage (+ 18 136) "4.5. Adjunctions, limits, and colimits")
% (find-books "__cats/__cats.el" "awodey")
% (find-awodeyctpage (+ 10 197) "9.6 RAPL")
\newpage
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