Here are some more names for stable functors, in addition to those provided by Paul Taylor. I believe that they first appeared in a paper by J.J. Kaput (Ill.J. Math. 16 (1972) 86-94) under the name "locally (left) adjunctable functor". I used "locally right adjoint functor" in my paper with Borger (Manuscripta Math. 19 (1976) 19-45) and in the more comprehensive paper (Comment. Math. Univ. St. Pauli 28 (1979) 179- 202). Both papers deal with the particular case of "locally reflective subcategories" for which there are new results to be found in a recent paper by Adamek and Volger ("On locally reflective categories of struct- ures"). Walter Tholen =====================================================================
I don't much like the locally as suggested by Walter, but it is probably too late to do much about it. I prefer multi-adjoint. Locally P means that P holds in every slice. Mike =====================================================================
Here is another contribution to the baptism of "stable functors". When a functor F represents a datatype construction, such as lists, trees or matrices, then it typically preserves pullbacks (in fact, all connected limits) and when applied to the terminal object 1 yields the "basic shape" F1 whose "elements" are the shapes of general data. For example, the list functor applied to 1 is a NNO since the shape of a list is its length. In this setting it is attractive to call pullback-preserving functors "shapely". Similarly, the natural transformations representing the generic operations on the datatypes (e.g. the appending of lists or flattening of trees) often have the property that the commuting squares arising in their definition are actually pullbacks. Such natural transformations are commonly known as cartesian natural transformations, but in this setting could also be called "shapely natural transformations". Thus there is a 2-category consisting of categories with finite (connected?) limits, shapely functors and shapely natural transformations, which is convenient for studying the general theory of datatypes. Barry Jay =====================================================================
No problem with Mike's remark that "locally P" should mean "P holds in every slice". This is for a property of a category. When using "locally right adjoint" almost two decades ago I asked myself whether the func- tor A/a --> B/Fa induced by F:A-->B (and an A-object a) is an accep- table notion of "slice of the functor F", decided that would be okay, and therefore used the name. Diers' notion of multi-adjointness gives more than just right adjointness of the slices A/a --> B/Fa : there is a smallness condition on the number of non-isomorphic "local units", and these are F-epic. Hence it would be confusing to use the name in a more general sense now. I may be wrong in assuming that the notion of "slice of a functor" as above is acceptable. Comments welcome. Walter Tholen. =====================================================================
I guess Walter is right. At least in the case of an initial object, it is the case that having a multi-initial object is having an initial object in each slice. What I hadn't thought through was the fact that if a category has a multi-initial object and has a terminal object, then it has an initial object. So having an initial object in each slice does not imply having an initial object (whereas for most properties P, locally P implies P). There is a minor problem with duality. While it is true that a category with a multi-terminal object has a terminal object in each slice, that is also true of a category without a multi-terminal object. So you want that for every co-slice. Michael =====================================================================
participants (4)
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barr＠triples.Math.McGill.CA -
cbj＠dcs.edinburgh.ac.UK -
Walter -
Walter Tholen