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Friday, April 17, 2020

Kelly ECT displayed formulae and diagrams

PRELIMINARY DRAFT --- CUSTOMIZED, NON-STANDARD

This is just a beginning on that project. 
Currently only a few selected formulae and diagt are listed. Also some of the Kelly's words have been included.
The formulae are not exactly what Kelly wrote in ECT. I have transformed them somewhat. 

AAATAequalizerAAATAρσA,AA[AAA,ATA]λAprojAATA
[X,AAATA][X,λA][X,ATA] makes its source the end of lts targets, inducing[X,AAATA]AA[X,ATA]
(A,B)ABA,BTA,BBBAAA,BTA,B
T[A,B]SAAATBASequalizerAAATBASρσA,AA[AAA,ATBAS]EAprojAATBAS
[X,T[A,V]S]T[A,V][X,S]

Given a V-functor F:AV and an object K of A as in 1.9, 
we have the map (arrow in V0) KFA:KAA[FK,FA]
which is.V-natural in A by 1.8(b).
The transform ϕA:FK[KAA,FA] of KFA under the (symmetry) adjunction XV0[Y,Z]YV0[X,Z]
is (V-natural in A) by 1.8(m).
(The stronger Yoneda lemma) is 
the assertion that (2.30) expresses FK as the end A[KAA,FA]
so that we have an isomorphism (natural in K) ϕ:FKKA[A,V]F.
FK was a single-variable functor; Kelly generalizes to a functor CPK of two variables in the (natural in the "extra" variable C, as well as in K):
ϕ=CϕK:CPKKA[A,V]CP

BB{KK,G}(3.1)KK[K,V]BBG(KKG)BB(3.5)KK[Kop,V]GBB

\begin{align} \hom C Q {\{F,G\}} & \mathop\cong\limits^{Q \text{ preserves}}_{\text{limit}} & {\{F,\hom C Q G\}} & \mathop\cong\limits^{ (3.7)} & \hom F {[\calK,\calV]} {\hom C Q G} \\ \llap{\text{defn of $T$ }} {\wr\Vert} \\ \boxed{\hom C \calC {\{F,G\}T}} &&&& {\wr\Vert} \rlap{\text{ defn of $T$}}  \\ \\ \boxed{\hom C \calC {\{F,GT\}}} && \mathop\cong\limits^{(3.1)} && \hom F {[\calK,\calV]} {\hom C \calC {GT}} \tag{3.15} \end{align}

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