On Fri, 4 Jan 2002 jdolan@math.ucr.edu wrote:
in lambek and scott's "introduction to higher order categorical logic", in the "historical comments" on section 7 of part 1, on page 116, they say:
"it should be emphasized that, as long as equalizers are excluded from the definition of cartesian closed categories, adjoining an indeterminate of type a is not the same as forming the slice category c/a, but it is once equalizers are included. the latter was observed by grothendieck and joyal (see part 2, section 16, exercise 2)."
i've been having a bit of trouble trying to understand exactly what they mean here. as far as i can tell, the mentioned "exercise 2" concerns the case where the category c is a topos, but the above quoted statement makes it sound like the equivalence between adjoining an indeterminate and forming a slice category should hold in the case where c is just a "cartesian closed category with equalizers", or something like that. offhand though i couldn't think of a way to get a result along these lines to be true. for one thing, the assumption that c has finite limits and exponentials doesn't even seem to compel the slice category c/a to have exponentials. (i think the category of co-commutative co-algebras over a field provides a counterexample.)
can someone explain where i'm making a mistake here? or is it just that lambek and scott were only referring to the topos case, as discussed in their "exercise 2"?
Only Lambek and Scott can say what they meant by this remark. However, the result is true for arbitrary locally cartesian closed categories (i.e., if C is a lccc, then C/a is the free lccc-with-an-indeterminate- of-type-a generated by C); it doesn't need the extra structure of a topos. It is, of course, not true for `merely' cartesian closed categories with equalizers, precisely because under these hypotheses C/a needn't be cartesian closed. Peter Johnstone