categories: Re: adjoining an indeterminate

["P. Scott" <scpsg@matrix.cc.uottawa.ca>]

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)."
>
..............................

The remark was a bit loosely worded, but of course the exercise
in question relates to the fact that in toposes (and still more 
generally in doctrines of ccc's  with finite limits and for which
slicing is well-behaved, e.g. locally ccc's) then one
may interpret slicing as "adjoining an indeterminate".  This is 
not the case for ordinary ccc's (no finite limits).  For further
info, see the exercises of L&S, p. 64.

The reason for making the remark in the first place was that when
Lambek and I used to lecture on ccc's long ago, and we used the word
"indeterminate", someone from the audience would invariably say
"oh, you mean take the slice category" and a long discussion would
then ensue...

				Cheers,
				Phil Scott