Re: locally cartesian closed categories

[Lutz Schroeder <Lutz.Schroeder@dfki.de>]

> I agree with Jean Benabou, Fred Linton and Vaughan Pratt that the
> definition of a locally cartesian closed category should NOT require
> a terminal object.   

[...]

> I confess that I'm a bit surprised to find that the consensus agrees
> with me, so to set matters straight I should also point out that my
> argument applies equally to elementary toposes and other familiar
> structures of categorical logic.

Such as cartesian closed categories, for instance. I would like to take
the opportunity to point to my paper "Life without the terminal type" in
CSL 2001, where I prove that every "almost" cartesian category, i.e. one
without a terminal object, extends uniquely to a cartesian closed
category with terminal object. There is also a similar result for
toposes; the wording is not quite as straightforward as for cartesian
closed categories, as one has to formulate (say) the definition of a
subobject classifier without reference to a global element True. I
recall having thought about locally cartesian closed categories as well,
but I do not think I really got anywhere (and actually I just see
there's a remark in the paper that says as much).

Lutz Schröder

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