Date: Thu, 28 Oct 1993 21:30:45 -0700 From: "William H. Rowan" <rowan@crl.com> What do you call it if you have a category C, and you have a class X of arrows of C such that if x in X, then gxf in X for all composable isomorphisms f and g. This rolls connectedness and abstractness into the one condition, as follows. It had therefore better have some independently justifiable redeeming social value before burdening math dictionaries with its own name. 1. Dropping "iso" from your condition strengthens it to the notion of *connected component*. 2. Requiring that X be closed under all automorphisms F:C->C, in the sense that x in X implies F(x) in X, strengthens your condition to what might reasonably be called an *abstract class*. A category with two distinct isomorphic connected components (e.g. *-->* *-->*) witnesses the strictness of this strengthening, in that a single component does not form an abstract class but does satisfy your condition. Therefore, as a strong common weakening of "connected component" and "abstract class" (but not the strongest, being strictly weaker than their disjunction), it would seem that your condition deserves nothing shorter than "connected-abstract class." A functor C->C' which is one leg of an equivalence takes such a set to another one X' in C', [...] No, F(X) need not be a connected-abstract class even if we assume of X not the disjunction but the conjunction, that X is *both* a connected component and an abstract class. Witness any full embedding F:G->H of a group G (as a one-object groupoid) in a connected groupoid H having more than one object. Here F is an equivalence, and the set X of all morphisms of G is both a connected component and an abstract class. But not only is F(X) neither a connected component nor an abstract class of H, it is not even a connected-abstract class of H. What interesting theorem justifies adding "connected-abstract" to the lexicon? -- Vaughan Pratt (FTPables: boole.stanford.edu:/pub/ABSTRACTS.) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

What do you call it if you have a category C, and you have a class X of arrows of C such that if x in X, then gxf in X for all composable isomorphisms f and g?

I haven't seen a name, but can I suggest calling X a "2-sided sieve"? (Or "2-sided crible"?) Conceivably, people who know what a sieve is could hazzard a guess at what "2-sided sieve" means. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

This rolls connectedness and abstractness into the one condition, as follows. It had therefore better have some independently justifiable redeeming social value before burdening math dictionaries with its own name.

A category is a ring with many objects but no additive structure, and a presheaf is then a left module. A representable presheaf is the ring considered as a left module over itself (but there are lots of them because of the many objects), and a sieve - a subpresheaf of a representable presheaf - is a left ideal. The category is also a - just one - bimodule (or profunctor) over itself just as a ring is, and the concept under discusion, a subbimodule, is an ideal exactly as Max Kelly said (a 2-sided ideal, or 2-sided sieve). That's a justification by analogy with ring theory, though there is a gap: ideals of rings are good not just because they are subbimodules. They are also kernels of quotients, and once the additive structure is dropped then it is no longer true that quotients are equivalent to subbimodules. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++