large integrals via manageable ends?

large integrals via manageable ends?

[rjwood]

With reference to the following diagram in \CAT, where \set is the category
of small sets:

        F
  \A------->\B
   |
  P|
   |
   v
\set

it is classical that if \A is small and \B is locally small then the
left Kan extension of P along F exists, call it L, and for B in \B
LB=\int^A \B(FA,B) x PA. Clearly, local smallness of B can be weakened
to the requirement that all \B(FA,B) are small, a condition called
`admissibility' of F, by Street and Walters in their `Yoneda structures'
paper. The point is that a small integral of small sets is small.

However, I claim that smallness of \A can be `weakened' to local smallness
of \set^{\A\op}. If we include \set in a category \SET of sets large
enough to contain all the LB as above, via I:\set----->\SET then the
description
of L also gives a left Kan extension of IP along F. Now write p(LB) for
the power set of LB and consider the following calculation:

p(LB)=\SET(\int^A \B(FA,B) x PA, 2)=\int_A\SET(\B(FA,B), 2^{PA})
      =\int_A\set(\B(FA,B), 2^{PA})
      =\set^{\A\op}(\B(F-,B),\set(P-,2))

It shows that local smallness of \set^{\A\op} implies p(LB) is small and
hence that LB is small. Thus for \set^{\A\op} locally small and F
admissible, the left Kan extension of any P:\A--->\set along F exists and
is given by the usual formula.

This calculation was prompted by questions similar to an open problem
in the paper by Freyd and Street `On the size of categories' in TAC V1.
Can anybody provide pointers to similar calculations where a big integral
is tamed by a manageable end?
Thanks, RJ Wood



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