Paul Taylor asks for a category in which REGULAR EPIS DO NOT COMPOSE. The short answer is |Cat . (Where |x denotes boldface-x, look at the canonical presentation, which has length two, of |3 in terms of free categories over sets, i.e., coproducts of copies of |2. The intervening algebras in this presentation are free categories over graphs.) In slightly greater generality, how do we produce examples of categories in which there are chains of length n of regular epis -- whose composites cannot be written as composites of fewer than n regular epis (for each n > 1) ? When I worried about this several decades ago, Peter Freyd gave an answer, in his essentially equationally defined categories [e.g., in the early pages of his Aspects of Topoi] -- defined by equations on a set of partial as well as entire operations -- in which the operations can be ordered so that for each partial operation its domain is defined by equations in earlier operations. |Cat is an example when n = 2. This leads into the subject of adjoint towers [Applegate-Tierney, LNM 137; Adamek-Herrlich-Rosicky c.1987] To get a tower of height n over |Sets^m (for any m ) there must be n batches of progressively "higher" operations, each requiring the last of the preceding batches, in the definitions of its domains. For the |Cat example, the intervening category is directed |Graphs. The base category is respectively |Sets or |Sets^2 , when we use the usual one-sorted or two-sorted definition of "category". And if we define "category" in terms of (commuting) "triangle", in place of "morphism", then the tower degenerates -- after the pattern of the main theorem of Gabriel-Ulmer [LNM 195 and LNM 221]. Now, if |Cat is interesting in this way, what does it mean that |Cat is a topos wrt categories enriched over it [as remarked by Ross Street several months ago here on the catnet], and what modifications are we going to be compelled to make when we attempt to carry topos-style thinking from |Sets over to |Cat? Talk about WHILE programs suggests talk about imperative languages. Perhaps more important, even, than their dealing with STATE is the way imperative languages can be explicit about extending or shrinking storage (using pointers). Manes and Moggi have shown us how monads can be used to model STATE and given us a category theoretic translation of the distinction between call-by-value and call-by-name [begin with Moggi's paper on theory.doc.ic.ac.uk]. Is there also a 2-category theoretic translation of the distinction between call-by-value and call-by-reference, that depends on the difference between the two kinds of composition we have in 2-categories? Here I doubt if the base 2-category can be |Cat. But |Cat (x) |Cat remains a possibility, where (x) denotes Gray's assymetric tensor product [LNM 391 and in the Heller-Tierney (ed.) Eilenberg festschrift]. Art Stone +++++++++++++++++++++++++++++++++++++++++++