Categorical, i have a question about a presentation of sets. The motivation for this question comes from a conversation i had a few years ago with Jamie Gabbay concerning the use of FM-set theory in semantics for 'freshness'. i was arguing that a certain use of 'reflection' gave a canonical (and smallest) set of atoms. Jamie objected that atoms are not allowed to have internal structure in FM-set theory. My question has to do with my response. Because my question has to do with a formulation in categorical language, rather than formulate it in that language, i will formulate it naively. In my view, heavily influenced by computing as it is, i see the basics of set theory as providing some operations for constructing and inspecting, de-structing a data type called set. Very primitively, we have operations for - extensionally constructing sets, '{ ... }' and - operations for intensionally constructing sets '{ ... | ... }' - operations for inspecting sets 'x in ... ' In this view, nothing prevents me from imagining two different versions of this data type. One of which i will call the 'black' version and one of which i will call the 'red' version. Initially, i might imagine these data types as copies of each other; but, we can only construct and inspect 'black sets' with 'black' braces and 'black' in predicate; and likewise for the 'red sets'. So, never the twain shall meet. Now, once we've built such a structure, there's nothing to prevent us from imaginging that the 'atoms' of a 'black' FM-set theory are none other than 'red sets'. Symmetrically, nothing prevents us from imagining that the 'atoms' of a 'red' FM-set theory are none other than 'black sets'. A suggestive use of data type specifications might illustrate the idea - Ordinary sets - Set ::= '{' Set* '}' - Red/black sets - BlackSet ::= '{b|' (BlackSet + RedAtom)* '|b}' - RedSet ::= '{r|' (RedSet + BlackAtom)* '|r}' - RedAtom ::= RedSet - BlackAtom ::= BlackSet Now, my question: is there a topos theoretic characterization of the obvious zoology that results from these musings? Best wishes, --greg -- L.G. Meredith Managing Partner Biosimilarity LLC 806 55th St NE Seattle, WA 98105 +1 206.650.3740 http://biosimilarity.blogspot.com