Algebraically compact functors

In a previous paper, we investigated the relation between the initial algebra and terminal coalgebra for an endofunctor on the category of sets. In this one we study conditions on a functor to be algebraically compact, which means that the canonical comparison morphism between those objects is an isomorphism. Introduction Suppose C is a category and T : C −→ C is a functor. In both [Barr, 1991] and [Freyd, 1991] it is shown that there is a canonical arrow between the initial T -algebra and terminal T -coalgebra and both papers study its properties in some special cases. Freyd has introduced the term algebraically compact to describe a category for which that arrow is always an isomorphism. He does not actually exhibit any non-trivial examples of such categories, although he claims that the realizable topos has a “small full reflective subcategory that is algebraically compact in the relevant sense, that is, the condition holds for every endofunctor that is definable as a functor in the topos.” This suggests that it might be worth restricting attention to functors that are “relevant”. For example, when dealing with categories enriched over some base category, it may be relevant to restrict to functors that preserve that enriched structure. For these and other reasons, we define a functor to be algebraically compact if the canonical map is an isomorphism. Freyd also defined a category to be algebraically complete if every functor has an initial algebra. Clearly an algebraically compact category is also algebraically complete. However we wish to explore a condition closely related to algebraic compactness that is meaningful even in the nonalgebraically-complete case. ∗ In the preparation of this paper, I have been assisted by grants from the NSERC of Canada and the FCAR du Québec. If T is an endofunctor, let us say that a fixed object for T is an object C with an isomorphism TC −→ C . This is a special kind of T algebra and, using the inverse isomorphism, it is also a special kind of T -coalgebra. An initial algebra, if one exists, is a fixed object and the initial fixed object and a terminal coalgebra, if one exists, is a fixed object and the terminal fixed object. One of the main interests is in the category of fixed objects. A functor need not have any fixed object. For example the covariant power set functor on the category of sets does not have any. In general, not very many categories are algebraically compact. However, it may happen that every functor in some usefully large class of functors is algebraically compact. For example, the homsets might be ordered and we may restrict to functors that preserve the order. In that case, we say that that class of functors is algebraically compact. Finally, we define a class of functors to be conditionally algebraically compact if every functor in the class that has a fixed object is algebraically compact. For various reasons, it appears that the category of CPOs (defined below) would prove to be a good source of examples. In fact, it is there that many of the models of invariant objects are found. And indeed we find a class of functors both on the that category and on the category of CPOs with bottom which are algebraically compact (Theorems 4.6 and 4.8). A CPO is a partially ordered set in which every directed set of elements has a sup. It is equivalent (using the axiom of choice) to suppose that every ordinal indexed increasing chain of elements has a sup. Among the motivations for Freyd’s paper was the feeling (which I shared) that there was something ad hoc about the embedding/projection pairs that have been used to find invariant objects for functors that were contravariant or of mixed variance. (See [Smith & Plotkin, 1983] or [Barr & Wells, 1990] for an explanation.) It was thus of considerable surprise to me to find embedding/projection sequences arising naturally in this investigation. In retrospect, it perhaps should not have been so surprising. Among the results found in Freyd’s paper are that invariant objects for covariant, contravariant and mixed variance functors are found under the same conditions. Originally, embedding/projection sequences were introduced to make variance irrelevant. In Freyd’s treatment, all functors are converted to covariant endofunctors on an appropriate category. The price to be paid is that now one needs not just an initial fixed point, but a simultaneously initial and terminal fixed point. The search for this turns out to lead quite naturally to embedding/projection sequences. It may well be that these sequences are inevitable in this connection, rather than just being a feature of one way of looking at it.