Subject: categories: my earlier comments on certain limits in algebras for a 2-category.

From: [email-masked] (Max Kelly)
Date: Tue, 2 Feb 1999 00:25:26 +1100 (EST)

After returning home and checking my references, I see that in [Blackwell,
Kelly, Power] we show that, for a 2-monad T on a complete 2-category K, the
category T-Alg, whose morphisms are the non-strict ones (the "to within a
coherent isomorphism" ones), admits products, inserters, and equifiers,
formed as in K, and hence such limits as follow from these, including the
cotensor product (-)^X for an X in K. What I said today about admitting all
FLEXIBLE limits is false; I had misremembered, and there is a counter-
-example in Example 6.2 of [Bird, Kelly, Power, Street, Flexible limits for
2-categories, JPAA 61 (1989), 1 - 27]. The flexibles would follow from those
above if idempotents were to split - but they don't in general.

I am strongly convinced that the above is true. The mere CATEGORY Ccc of
cartesian closed categories was shown to be monadic over the mere category
Cato of categories by Burroni, and this was checked and restated by Dubuc
and Kelly in [J. Algebra 81 (1983), 420 - 433]. Blackwell, Kelly, and Power
claimed on p.39 of [B,K,P] that Ccc, as a 2-category with only invertible
2-cells, is monadic (2-monadic, if you want the repeated emphasis) over the
complete 2-category Catg which is obtained from the 2-category Cat by
discarding all the non-invertible 2-cells. It is most unlikely that Ccc is
2-monadic over the usual 2-category Cat; for already symmetric monoidal
closed categories is known not to be so; see p.34 of [B,K,P].

I had strongly believed that Ccc was indeed 2-monadic over Catg; but we never
checked every detail of this particular example. Today my belief has been
shaken by two letters (one from Barry Jay) claiming that the monadicity of 
Ccc is false, unless the cartesian closed categories are supposed to admit
pullbacks. I shall try to check this out in the coming days - we were quite
sure that the structure is equational even without pullbacks, but perhaps we
were wrong. Certainly Burroni (see [Dubuc-Kelly for the reference) supposed
pullbacks present, because he was headed for toposes.

I must say that the letters I received today were imprecise about which
base 2-category was involved; to be fair, they were not really about 
monadicity but about the existence of A^2. However I formed theimpression
that the 2-categorical investigations in the references above have not (yet?)
become part of the common language of our colleagues. Note that the T-Alg
where these limits are to exist does NOT have the strict maps; so that it is
certainly not monadic in any classical sense. Clearly one cannot begin even
to pose such questions precisely without using 2-categorical notions. Let me
finish by saying that the universal property of A^2 is 2-categorical by its
very nature (if we are speaking of A^2 in T-alg, not just in Cat).

Oh, that's it! I have just seen what the point is. [B,K,P] is quite right in
asserting that Ccc has the limit A^2 formed as in Catg; but the limit A^2 in
Catg is not at all that in Cat; it is the category such that to give
B --> A^2 is to give two maps B --> A and a 2-cell between them; but the
2-cells in Catg are just the INVERTIBLE natural transformations, so that
A^2 in Catg is A^I in Cat, where I is the free isomorphism, not the free a
arrow 2.

So, all right, I give in. The [B,K,P] results don't give a cartesian closed
structure on the USUAL A^2 in Cat. Anyway, I've learned (or relearned)
something from thinking this through, and hope some others may benefit from
seeing these notions brought up.

And so to bed. Max Kelly.