Re: Confused about Cartesian closed categories
Justin Pearson asks:
Some books ... say that a CCC has a finite limits plus ... exponentials. ... other books seem to define it as finite products plus exponentials. ... can you get finite limits from finite products ... ?
Just as Eilenberg's "co-six" category (all sets / or all finite sets / of cardinality other than 6 ) provides an example of a closed category that's not (but is almost) cartesian closed, so what one might call "co-five" (all sets / or all finite sets / of cardinality other than 5 ) provides an example of a cartesian closed category (in the second sense) not having all equalizers (so not finitely complete). For some purposes one cares not a fig about the presence/absence of finite limits (other than products, of course) -- for other purposes one cares a lot. [Of course almost any prime could have been used in place of the prime 5 .] Fred E.J. Linton Wesleyan U. Math. Dept. 649 Sci. Tower Middletown, CT 06459 E-mail: <FLINTON@eagle.Wesleyan.EDU> ( or <fejlinton@{att|mci}mail.com> ) Tel.: + 1 203 776 2210 (home) or + 1 203 347 9411 x2249 (work)
I am afraid that different authors make different asssumptions. I guess the minimum you want is finite products (except that a C-monoid is sometimes defined as a CCC with one object, so it has only finite non-empty products), while others will want finite limits. They are definitely different, no doubt of that and the complexity of the computations in the initial CCC of those types is different too, so the difference is important. There is an open question of some interest: can an FP CCC with NNO be embedded into an FL CCC with NNO in a way that preserves the NNO. I conjecture that it cannot, but if it could, that would have some interesting consequences. --Michael
Justin Pearson asks what is the "correct" definition of Cartesian closed category. I would say that the definition most commonly used is that a category C is a cartesian closed category (ccc) if it has specified terminal object, binary products, and exponentials for each pair of objects. This means that there is a (natural and) precise correspondence between cccs, and equational theories with unit type, binary products and function types. What is potentially confusing is that a Cartesian category is often taken to mean a category with finite limits (binary products and equalisers) because Descartes's work involved equationally defined subsets. For this reason it might be best to call cccs (def. as having finite products and exponentials) by a different name, say exponential categories---but here "best" is clearly of dubious meaning ! Roy Crole.
participants (3)
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Fred E.J. Linton -
Michael Barr -
Roy Crole